| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 2·13-s − 14-s + 16-s − 6·17-s + 2·19-s − 23-s − 5·25-s − 2·26-s + 28-s − 31-s − 32-s + 6·34-s − 4·37-s − 2·38-s − 9·41-s + 11·43-s + 46-s + 12·47-s + 49-s + 5·50-s + 2·52-s + 9·53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.208·23-s − 25-s − 0.392·26-s + 0.188·28-s − 0.179·31-s − 0.176·32-s + 1.02·34-s − 0.657·37-s − 0.324·38-s − 1.40·41-s + 1.67·43-s + 0.147·46-s + 1.75·47-s + 1/7·49-s + 0.707·50-s + 0.277·52-s + 1.23·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350658 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389319151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389319151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.40131369941846, −11.91845350418425, −11.67606727655768, −11.03976819303612, −10.71069885782193, −10.42670392256093, −9.729476961422410, −9.264935866663970, −8.980297046091148, −8.325879786227421, −8.208963217555485, −7.326027396610771, −7.250396579678336, −6.569530644798743, −6.099394320885397, −5.597777626120121, −5.122511297062140, −4.443401904820973, −3.879709275557678, −3.572133247452819, −2.609537713213236, −2.286884257114542, −1.708191120011868, −1.052168417023677, −0.3734948754870208,
0.3734948754870208, 1.052168417023677, 1.708191120011868, 2.286884257114542, 2.609537713213236, 3.572133247452819, 3.879709275557678, 4.443401904820973, 5.122511297062140, 5.597777626120121, 6.099394320885397, 6.569530644798743, 7.250396579678336, 7.326027396610771, 8.208963217555485, 8.325879786227421, 8.980297046091148, 9.264935866663970, 9.729476961422410, 10.42670392256093, 10.71069885782193, 11.03976819303612, 11.67606727655768, 11.91845350418425, 12.40131369941846
