| L(s) = 1 | − 3.05·3-s + 0.773·5-s − 3.93·7-s + 6.35·9-s + 0.488·11-s + 1.20·13-s − 2.36·15-s + 1.22·17-s − 5.48·19-s + 12.0·21-s + 3.02·23-s − 4.40·25-s − 10.2·27-s + 3.71·29-s − 6.08·31-s − 1.49·33-s − 3.04·35-s + 9.95·37-s − 3.67·39-s + 0.904·41-s + 0.153·43-s + 4.91·45-s + 12.7·47-s + 8.49·49-s − 3.74·51-s − 13.1·53-s + 0.377·55-s + ⋯ |
| L(s) = 1 | − 1.76·3-s + 0.345·5-s − 1.48·7-s + 2.11·9-s + 0.147·11-s + 0.333·13-s − 0.610·15-s + 0.297·17-s − 1.25·19-s + 2.62·21-s + 0.631·23-s − 0.880·25-s − 1.97·27-s + 0.689·29-s − 1.09·31-s − 0.259·33-s − 0.514·35-s + 1.63·37-s − 0.588·39-s + 0.141·41-s + 0.0234·43-s + 0.733·45-s + 1.86·47-s + 1.21·49-s − 0.524·51-s − 1.81·53-s + 0.0508·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 3.05T + 3T^{2} \) |
| 5 | \( 1 - 0.773T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 11 | \( 1 - 0.488T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 + 6.08T + 31T^{2} \) |
| 37 | \( 1 - 9.95T + 37T^{2} \) |
| 41 | \( 1 - 0.904T + 41T^{2} \) |
| 43 | \( 1 - 0.153T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 8.23T + 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 4.65T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 1.53T + 89T^{2} \) |
| 97 | \( 1 - 0.103T + 97T^{2} \) |
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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
−7.88788486706324485452893849309, −6.96431732979562017458169132739, −6.37433942660362950204357452591, −6.02851688496953605614187180791, −5.32445008854165775775627066675, −4.35696590058572645509892811427, −3.63567625992955794552654573726, −2.37620147767412723926429571467, −1.02576514802809954639760642376, 0,
1.02576514802809954639760642376, 2.37620147767412723926429571467, 3.63567625992955794552654573726, 4.35696590058572645509892811427, 5.32445008854165775775627066675, 6.02851688496953605614187180791, 6.37433942660362950204357452591, 6.96431732979562017458169132739, 7.88788486706324485452893849309
