| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3·7-s − 8-s + 9-s + 6.27·11-s − 12-s + 4.27·13-s + 3·14-s + 16-s − 5.27·17-s − 18-s − 4.27·19-s + 3·21-s − 6.27·22-s + 23-s + 24-s − 4.27·26-s − 27-s − 3·28-s + 5.54·29-s + 6·31-s − 32-s − 6.27·33-s + 5.27·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 0.333·9-s + 1.89·11-s − 0.288·12-s + 1.18·13-s + 0.801·14-s + 0.250·16-s − 1.27·17-s − 0.235·18-s − 0.980·19-s + 0.654·21-s − 1.33·22-s + 0.208·23-s + 0.204·24-s − 0.838·26-s − 0.192·27-s − 0.566·28-s + 1.03·29-s + 1.07·31-s − 0.176·32-s − 1.09·33-s + 0.904·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9816978743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9816978743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 6.27T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 29 | \( 1 - 5.54T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 - 0.274T + 43T^{2} \) |
| 47 | \( 1 + 0.725T + 47T^{2} \) |
| 53 | \( 1 + 4.54T + 53T^{2} \) |
| 59 | \( 1 + 2.54T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 0.450T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 0.725T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.801012976646974313941080866245, −7.989208059748049829370859807515, −6.68084775328972205910860537781, −6.43159868266270945459865001299, −6.20640679644333468405607633455, −4.62826327387235386446208649450, −3.93654045213171685287709162609, −3.00072267742670920775548134199, −1.70482563237124050045064784887, −0.69274100054239107301987565343,
0.69274100054239107301987565343, 1.70482563237124050045064784887, 3.00072267742670920775548134199, 3.93654045213171685287709162609, 4.62826327387235386446208649450, 6.20640679644333468405607633455, 6.43159868266270945459865001299, 6.68084775328972205910860537781, 7.989208059748049829370859807515, 8.801012976646974313941080866245
