| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s − 2·12-s − 4·13-s − 14-s + 16-s − 6·17-s + 18-s + 2·19-s + 2·21-s − 2·24-s − 5·25-s − 4·26-s + 4·27-s − 28-s + 6·29-s + 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.408·24-s − 25-s − 0.784·26-s + 0.769·27-s − 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52094 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52094 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7610545264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7610545264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 61 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.39739856699375, −13.95412017912817, −13.34555675015517, −12.91669952593805, −12.35512799315394, −11.91580922343089, −11.47998133232571, −11.12962561797187, −10.45244768323902, −9.885104568845423, −9.595093155313694, −8.624205279424579, −8.173449077566698, −7.335661321548955, −6.862272720750796, −6.347283312112179, −6.015238162583719, −5.179006730499253, −4.854921469992446, −4.380802654301130, −3.565746484965917, −2.797457354068156, −2.279071975190648, −1.344304445236489, −0.2899210345445212,
0.2899210345445212, 1.344304445236489, 2.279071975190648, 2.797457354068156, 3.565746484965917, 4.380802654301130, 4.854921469992446, 5.179006730499253, 6.015238162583719, 6.347283312112179, 6.862272720750796, 7.335661321548955, 8.173449077566698, 8.624205279424579, 9.595093155313694, 9.885104568845423, 10.45244768323902, 11.12962561797187, 11.47998133232571, 11.91580922343089, 12.35512799315394, 12.91669952593805, 13.34555675015517, 13.95412017912817, 14.39739856699375
