| L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s − 3·8-s + 9-s + 2·10-s + 6.24·11-s + 12-s − 0.828·13-s − 2·15-s − 16-s + 0.828·17-s + 18-s + 19-s − 2·20-s + 6.24·22-s − 2.24·23-s + 3·24-s − 25-s − 0.828·26-s − 27-s − 1.65·29-s − 2·30-s + 0.828·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.5·4-s + 0.894·5-s − 0.408·6-s − 1.06·8-s + 0.333·9-s + 0.632·10-s + 1.88·11-s + 0.288·12-s − 0.229·13-s − 0.516·15-s − 0.250·16-s + 0.200·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s + 1.33·22-s − 0.467·23-s + 0.612·24-s − 0.200·25-s − 0.162·26-s − 0.192·27-s − 0.307·29-s − 0.365·30-s + 0.148·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.277871513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.277871513\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 + 1.65T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
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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
−9.060706973614027918032371588822, −8.085401829052476169004605324907, −6.94920502701385306542797534032, −6.17695662403297948521129459809, −5.83203985676142917041453773311, −4.89925993194376337297409156597, −4.16137327840218980552045591828, −3.42665088049623629573086463626, −2.07835637568624971126478106384, −0.908061193751690879881029934969,
0.908061193751690879881029934969, 2.07835637568624971126478106384, 3.42665088049623629573086463626, 4.16137327840218980552045591828, 4.89925993194376337297409156597, 5.83203985676142917041453773311, 6.17695662403297948521129459809, 6.94920502701385306542797534032, 8.085401829052476169004605324907, 9.060706973614027918032371588822
