| L(s) = 1 | − 3-s − 2.29·5-s + 7-s + 9-s − 2.78·11-s + 13-s + 2.29·15-s − 0.786·17-s + 0.297·19-s − 21-s + 1.87·23-s + 0.276·25-s − 27-s + 8.46·29-s − 1.51·31-s + 2.78·33-s − 2.29·35-s − 2.36·37-s − 39-s + 12.7·41-s + 0.297·43-s − 2.29·45-s − 2.48·47-s + 49-s + 0.786·51-s − 12.8·53-s + 6.40·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.02·5-s + 0.377·7-s + 0.333·9-s − 0.840·11-s + 0.277·13-s + 0.593·15-s − 0.190·17-s + 0.0681·19-s − 0.218·21-s + 0.390·23-s + 0.0553·25-s − 0.192·27-s + 1.57·29-s − 0.271·31-s + 0.485·33-s − 0.388·35-s − 0.388·37-s − 0.160·39-s + 1.99·41-s + 0.0453·43-s − 0.342·45-s − 0.363·47-s + 0.142·49-s + 0.110·51-s − 1.76·53-s + 0.863·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.29T + 5T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 17 | \( 1 + 0.786T + 17T^{2} \) |
| 19 | \( 1 - 0.297T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 - 0.297T + 43T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6.95T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 6.29T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.901383262077018788052571491217, −7.42095846807216717019769918438, −6.55058012124830463883913687360, −5.78513911558027241910164588009, −4.88633392686982527093012871025, −4.39664435473191843827778907189, −3.45115312084923236484079140066, −2.50692697155346258517701287262, −1.16541234298477770110832885668, 0,
1.16541234298477770110832885668, 2.50692697155346258517701287262, 3.45115312084923236484079140066, 4.39664435473191843827778907189, 4.88633392686982527093012871025, 5.78513911558027241910164588009, 6.55058012124830463883913687360, 7.42095846807216717019769918438, 7.901383262077018788052571491217
