| L(s) = 1 | + 34·9-s + 140·13-s + 112·17-s + 112·19-s + 20·25-s + 520·43-s − 952·47-s + 842·49-s − 576·53-s + 1.17e3·59-s + 240·67-s − 294·81-s + 2.85e3·83-s + 1.98e3·89-s + 1.59e3·101-s + 672·103-s + 4.76e3·117-s + 1.69e3·121-s + ⋯ |
| L(s) = 1 | + 1.25·9-s + 2.98·13-s + 1.59·17-s + 1.35·19-s + 4/25·25-s + 1.84·43-s − 2.95·47-s + 2.45·49-s − 1.49·53-s + 2.59·59-s + 0.437·67-s − 0.403·81-s + 3.77·83-s + 2.36·89-s + 1.57·101-s + 0.642·103-s + 3.76·117-s + 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.37745626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.37745626\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 17 | $D_{4}$ | \( 1 - 112 T + 382 p T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 1450 T^{4} - 34 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 4 p T^{2} + 12342 T^{4} - 4 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 842 T^{2} + 410922 T^{4} - 842 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 1698 T^{2} + 3550826 T^{4} - 1698 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 70 T + 4002 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 20346 T^{2} + 209323274 T^{4} - 20346 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 74036 T^{2} + 2514340758 T^{4} - 74036 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 114890 T^{2} + 5074759530 T^{4} - 114890 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 116372 T^{2} + 7903483062 T^{4} - 116372 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 158724 T^{2} + 15178105958 T^{4} - 158724 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 260 T + 128262 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 476 T + 257822 T^{2} + 476 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 288 T + 248662 T^{2} + 288 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 588 T + 335494 T^{2} - 588 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 425396 T^{2} + 97219598358 T^{4} - 425396 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 120 T + 571334 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 1379018 T^{2} + 731023514346 T^{4} - 1379018 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 55196 T^{2} + 301853502630 T^{4} + 55196 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 1282922 T^{2} + 849811865706 T^{4} - 1282922 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1428 T + 1595158 T^{2} - 1428 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 994 T + 1642394 T^{2} - 994 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 3375140 T^{2} + 4511732954310 T^{4} - 3375140 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.068657600064180299092164716480, −7.950475316337965870719207906950, −7.931976520781335138981494286659, −7.35689109916127315112995066043, −7.19272899340175634812227521762, −6.93670366358621845677797323506, −6.64697839752185359816785720403, −6.31656103451142006192781817718, −5.96085506865641306019120225892, −5.88806023522228958601623205952, −5.64719395424340608427659999785, −5.20914196008849948466424555207, −4.91086513169062701552069607786, −4.59960934722369966216956883390, −4.34436821062144865713286689474, −3.65007862120171194154545927916, −3.63859512478006888677454889118, −3.39798344538715395176162583128, −3.37162438896962760982434464865, −2.37065474905028486258482461663, −2.29668252585318136261265721065, −1.46126581980983879775679778368, −1.33535045370642008471297947863, −0.825916877904518928838301523312, −0.74055304948636410348533807081,
0.74055304948636410348533807081, 0.825916877904518928838301523312, 1.33535045370642008471297947863, 1.46126581980983879775679778368, 2.29668252585318136261265721065, 2.37065474905028486258482461663, 3.37162438896962760982434464865, 3.39798344538715395176162583128, 3.63859512478006888677454889118, 3.65007862120171194154545927916, 4.34436821062144865713286689474, 4.59960934722369966216956883390, 4.91086513169062701552069607786, 5.20914196008849948466424555207, 5.64719395424340608427659999785, 5.88806023522228958601623205952, 5.96085506865641306019120225892, 6.31656103451142006192781817718, 6.64697839752185359816785720403, 6.93670366358621845677797323506, 7.19272899340175634812227521762, 7.35689109916127315112995066043, 7.931976520781335138981494286659, 7.950475316337965870719207906950, 8.068657600064180299092164716480
