| L(s) = 1 | − 9-s − 6·19-s − 8·29-s − 14·31-s + 12·41-s + 13·49-s − 20·59-s − 2·61-s − 28·71-s − 16·79-s + 81-s − 12·101-s + 30·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 6·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.37·19-s − 1.48·29-s − 2.51·31-s + 1.87·41-s + 13/7·49-s − 2.60·59-s − 0.256·61-s − 3.32·71-s − 1.80·79-s + 1/9·81-s − 1.19·101-s + 2.87·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 0.458·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5714483967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5714483967\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−9.340910973639727101030373707159, −8.869210241208034633707728558690, −8.639769256121958995173049553935, −7.74570039723333436377873277899, −7.68276768189239902281564746574, −7.27930259799018092915697048342, −7.05054065531410228618352597696, −6.18381423301317560512408295484, −6.08917022798132104600834804322, −5.72223677903823098917009684268, −5.35210039435203600765445605149, −4.67419065467113868430514364543, −4.38214605879458415108493691353, −3.76211544891827657081617539049, −3.68823231709150720911723401967, −2.72292101756700228693606410788, −2.60224697650086629887193494503, −1.75387266958191001665915908677, −1.47272770460912329782683505230, −0.24592817794643946973030405775,
0.24592817794643946973030405775, 1.47272770460912329782683505230, 1.75387266958191001665915908677, 2.60224697650086629887193494503, 2.72292101756700228693606410788, 3.68823231709150720911723401967, 3.76211544891827657081617539049, 4.38214605879458415108493691353, 4.67419065467113868430514364543, 5.35210039435203600765445605149, 5.72223677903823098917009684268, 6.08917022798132104600834804322, 6.18381423301317560512408295484, 7.05054065531410228618352597696, 7.27930259799018092915697048342, 7.68276768189239902281564746574, 7.74570039723333436377873277899, 8.639769256121958995173049553935, 8.869210241208034633707728558690, 9.340910973639727101030373707159
