| L(s) = 1 | + 2-s + 4-s + 4·5-s − 7-s + 8-s + 4·10-s + 2·13-s − 14-s + 16-s − 17-s + 4·20-s + 4·23-s + 11·25-s + 2·26-s − 28-s + 8·29-s + 32-s − 34-s − 4·35-s + 4·37-s + 4·40-s − 2·41-s + 8·43-s + 4·46-s + 8·47-s + 49-s + 11·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s + 0.353·8-s + 1.26·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.894·20-s + 0.834·23-s + 11/5·25-s + 0.392·26-s − 0.188·28-s + 1.48·29-s + 0.176·32-s − 0.171·34-s − 0.676·35-s + 0.657·37-s + 0.632·40-s − 0.312·41-s + 1.21·43-s + 0.589·46-s + 1.16·47-s + 1/7·49-s + 1.55·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.334964779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.334964779\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.08105949710906, −12.34100847841125, −12.18154534733860, −11.43309502307719, −10.84829075041310, −10.56150075565572, −10.10482995930906, −9.660797815285436, −9.116403056868477, −8.741140912449056, −8.295435203772781, −7.308026516485494, −7.115954421102305, −6.438711620784485, −6.020595094016797, −5.842479983684143, −5.180059982549365, −4.707558849865552, −4.203106481011761, −3.455692917668005, −2.865389614580238, −2.472937154817748, −1.975459639358264, −1.167715272273262, −0.7915597107882038,
0.7915597107882038, 1.167715272273262, 1.975459639358264, 2.472937154817748, 2.865389614580238, 3.455692917668005, 4.203106481011761, 4.707558849865552, 5.180059982549365, 5.842479983684143, 6.020595094016797, 6.438711620784485, 7.115954421102305, 7.308026516485494, 8.295435203772781, 8.741140912449056, 9.116403056868477, 9.660797815285436, 10.10482995930906, 10.56150075565572, 10.84829075041310, 11.43309502307719, 12.18154534733860, 12.34100847841125, 13.08105949710906
