| L(s) = 1 | − 1.07·3-s + 3.27·5-s − 7-s − 1.83·9-s + 1.82·11-s − 1.36·13-s − 3.52·15-s − 7.22·17-s + 19-s + 1.07·21-s + 5.75·23-s + 5.71·25-s + 5.21·27-s + 3.58·29-s − 2.46·31-s − 1.96·33-s − 3.27·35-s − 6.01·37-s + 1.46·39-s − 4.40·41-s − 11.7·43-s − 6.01·45-s + 4.85·47-s + 49-s + 7.78·51-s + 8.43·53-s + 5.97·55-s + ⋯ |
| L(s) = 1 | − 0.622·3-s + 1.46·5-s − 0.377·7-s − 0.612·9-s + 0.550·11-s − 0.378·13-s − 0.910·15-s − 1.75·17-s + 0.229·19-s + 0.235·21-s + 1.19·23-s + 1.14·25-s + 1.00·27-s + 0.665·29-s − 0.442·31-s − 0.342·33-s − 0.553·35-s − 0.989·37-s + 0.235·39-s − 0.687·41-s − 1.79·43-s − 0.897·45-s + 0.708·47-s + 0.142·49-s + 1.09·51-s + 1.15·53-s + 0.805·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.07T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 7.22T + 17T^{2} \) |
| 23 | \( 1 - 5.75T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 + 6.01T + 37T^{2} \) |
| 41 | \( 1 + 4.40T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 4.85T + 47T^{2} \) |
| 53 | \( 1 - 8.43T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 9.39T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 2.21T + 79T^{2} \) |
| 83 | \( 1 - 8.86T + 83T^{2} \) |
| 89 | \( 1 + 4.69T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−6.93728522313854577237955679267, −6.73568326456259729232926006865, −6.14798650445148564861379224387, −5.21385615954914142643673899118, −5.08664285236055635572505297258, −3.91113212663094960189461653851, −2.86137015450365774285343082302, −2.23778311085979318087476709485, −1.27021511026695024072281988547, 0,
1.27021511026695024072281988547, 2.23778311085979318087476709485, 2.86137015450365774285343082302, 3.91113212663094960189461653851, 5.08664285236055635572505297258, 5.21385615954914142643673899118, 6.14798650445148564861379224387, 6.73568326456259729232926006865, 6.93728522313854577237955679267
