| L(s) = 1 | + 2-s − 11·4-s + 14·7-s − 15·8-s + 4·11-s + 22·13-s + 14·14-s + 61·16-s − 58·17-s + 4·22-s − 82·23-s + 22·26-s − 154·28-s + 334·29-s − 210·31-s + 89·32-s − 58·34-s − 6·37-s + 176·41-s − 46·43-s − 44·44-s − 82·46-s − 514·47-s + 147·49-s − 242·52-s − 808·53-s − 210·56-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.37·4-s + 0.755·7-s − 0.662·8-s + 0.109·11-s + 0.469·13-s + 0.267·14-s + 0.953·16-s − 0.827·17-s + 0.0387·22-s − 0.743·23-s + 0.165·26-s − 1.03·28-s + 2.13·29-s − 1.21·31-s + 0.491·32-s − 0.292·34-s − 0.0266·37-s + 0.670·41-s − 0.163·43-s − 0.150·44-s − 0.262·46-s − 1.59·47-s + 3/7·49-s − 0.645·52-s − 2.09·53-s − 0.501·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 2598 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22 T + 2458 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 58 T + 7794 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5490 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 82 T + 25182 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 334 T + 72842 T^{2} - 334 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 210 T + 69230 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 97490 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 176 T + 134094 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 46 T + 42430 T^{2} + 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 514 T + 269870 T^{2} + 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 808 T + 449478 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 284 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 618 T + 515018 T^{2} + 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 694 T + 681118 T^{2} + 694 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 814 T + 769934 T^{2} - 814 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 82 T + 422290 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 600 T + 618846 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 268 T + 779030 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 72 T + 448286 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1626 T + 1498938 T^{2} + 1626 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.868738706159550935696963911808, −8.447434868479384988725691843532, −8.147036793056954559523426040963, −7.86596886154450135500752780172, −7.35832138870063003989500672696, −6.70172929150530649428914671511, −6.29938672590132057834484863921, −6.11353470709332543919470748094, −5.25216445786932463049025752119, −5.14565052382689317365983629469, −4.59449175593685201203137836104, −4.43223108983375048192022938937, −3.71311493202852684103394189495, −3.59802994012841452569613876974, −2.75405815060199926597406437818, −2.27167902505665643537899197277, −1.42982863855845461109290695046, −1.12816888764171786518160287592, 0, 0,
1.12816888764171786518160287592, 1.42982863855845461109290695046, 2.27167902505665643537899197277, 2.75405815060199926597406437818, 3.59802994012841452569613876974, 3.71311493202852684103394189495, 4.43223108983375048192022938937, 4.59449175593685201203137836104, 5.14565052382689317365983629469, 5.25216445786932463049025752119, 6.11353470709332543919470748094, 6.29938672590132057834484863921, 6.70172929150530649428914671511, 7.35832138870063003989500672696, 7.86596886154450135500752780172, 8.147036793056954559523426040963, 8.447434868479384988725691843532, 8.868738706159550935696963911808
