| L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 4·11-s + 2·13-s − 4·14-s + 5·16-s − 2·17-s − 2·19-s + 8·22-s − 2·23-s − 4·26-s + 6·28-s + 2·31-s − 6·32-s + 4·34-s + 4·37-s + 4·38-s − 8·41-s + 6·43-s − 12·44-s + 4·46-s − 8·49-s + 6·52-s − 8·56-s + 6·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 1.20·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s − 0.485·17-s − 0.458·19-s + 1.70·22-s − 0.417·23-s − 0.784·26-s + 1.13·28-s + 0.359·31-s − 1.06·32-s + 0.685·34-s + 0.657·37-s + 0.648·38-s − 1.24·41-s + 0.914·43-s − 1.80·44-s + 0.589·46-s − 8/7·49-s + 0.832·52-s − 1.06·56-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 135 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 164 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 135 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 167 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.74125994588042306531232476338, −7.45951049788293498263211096513, −6.87176852707606962972417218936, −6.85887187534275611072641475834, −6.20841489097919530459648877439, −6.12538314503489766221221233957, −5.46955190622679789464938566091, −5.41497128270976921360655609535, −4.69346353739913181874572175716, −4.65671713790458360077584951458, −3.84457553767279559369535055902, −3.80527472715186443151093871115, −2.92328063462744528671562444541, −2.80462236326792919036270373649, −2.27033663724922039899855640776, −1.95314250417126341780329729659, −1.32259531930289380905985966900, −1.08569765668996990474122397801, 0, 0,
1.08569765668996990474122397801, 1.32259531930289380905985966900, 1.95314250417126341780329729659, 2.27033663724922039899855640776, 2.80462236326792919036270373649, 2.92328063462744528671562444541, 3.80527472715186443151093871115, 3.84457553767279559369535055902, 4.65671713790458360077584951458, 4.69346353739913181874572175716, 5.41497128270976921360655609535, 5.46955190622679789464938566091, 6.12538314503489766221221233957, 6.20841489097919530459648877439, 6.85887187534275611072641475834, 6.87176852707606962972417218936, 7.45951049788293498263211096513, 7.74125994588042306531232476338
