| L(s) = 1 | + 3·2-s + 6·3-s + 4-s + 18·6-s + 6·7-s + 3·8-s + 27·9-s − 42·11-s + 6·12-s + 78·13-s + 18·14-s + 9·16-s + 102·17-s + 81·18-s + 56·19-s + 36·21-s − 126·22-s − 48·23-s + 18·24-s + 234·26-s + 108·27-s + 6·28-s − 318·29-s + 52·31-s − 165·32-s − 252·33-s + 306·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1.15·3-s + 1/8·4-s + 1.22·6-s + 0.323·7-s + 0.132·8-s + 9-s − 1.15·11-s + 0.144·12-s + 1.66·13-s + 0.343·14-s + 9/64·16-s + 1.45·17-s + 1.06·18-s + 0.676·19-s + 0.374·21-s − 1.22·22-s − 0.435·23-s + 0.153·24-s + 1.76·26-s + 0.769·27-s + 0.0404·28-s − 2.03·29-s + 0.301·31-s − 0.911·32-s − 1.32·33-s + 1.54·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.855870013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.855870013\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + p^{3} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 326 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 42 T + 2734 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 p T + 5546 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 p T + 11402 T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 56 T + 8598 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 48 T + 24254 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 318 T + 55978 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 52 T + 58782 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 306 T + 94826 T^{2} - 306 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 408 T + 177982 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 120 T + 68150 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 180 T + 128990 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 402 T + 269234 T^{2} - 402 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 186 T + 419038 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 732 T + 698582 T^{2} - 732 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 36 T + 384046 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1332 T + 1102034 T^{2} - 1332 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 380 T + 99678 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 984 T + 942182 T^{2} + 984 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1116 T + 1508758 T^{2} - 1116 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 768 T + 1382402 T^{2} + 768 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.96421613736799455626654224012, −13.90911486129079565122183840125, −13.30445648441860509218390676394, −13.02869188257218953584293039039, −12.46361857535817110721507238677, −11.67176241449830026330259873699, −11.00006750586232230937605439277, −10.52267456278578779951991535147, −9.705511164404952700857363636706, −9.300029839835695070702623422304, −8.358410719490616120529746773823, −7.964995495075949724713933550451, −7.53332712345290363728509280646, −6.54672827298846704063399958730, −5.35242682104027895694792299706, −5.34165747804085571998374927723, −3.86442451056307274839247791522, −3.79684465948635158134660234263, −2.67789297042972100403261059468, −1.42782798213752939473729709287,
1.42782798213752939473729709287, 2.67789297042972100403261059468, 3.79684465948635158134660234263, 3.86442451056307274839247791522, 5.34165747804085571998374927723, 5.35242682104027895694792299706, 6.54672827298846704063399958730, 7.53332712345290363728509280646, 7.964995495075949724713933550451, 8.358410719490616120529746773823, 9.300029839835695070702623422304, 9.705511164404952700857363636706, 10.52267456278578779951991535147, 11.00006750586232230937605439277, 11.67176241449830026330259873699, 12.46361857535817110721507238677, 13.02869188257218953584293039039, 13.30445648441860509218390676394, 13.90911486129079565122183840125, 13.96421613736799455626654224012
