| L(s) = 1 | + 3·3-s − 7.29·5-s + 5.87·7-s + 9·9-s − 51.1·11-s − 13·13-s − 21.8·15-s − 73.7·17-s + 59.9·19-s + 17.6·21-s + 69.8·23-s − 71.8·25-s + 27·27-s − 294.·29-s − 334.·31-s − 153.·33-s − 42.8·35-s + 261.·37-s − 39·39-s + 222.·41-s + 79.2·43-s − 65.6·45-s − 584.·47-s − 308.·49-s − 221.·51-s + 465.·53-s + 373.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.652·5-s + 0.317·7-s + 0.333·9-s − 1.40·11-s − 0.277·13-s − 0.376·15-s − 1.05·17-s + 0.723·19-s + 0.183·21-s + 0.633·23-s − 0.574·25-s + 0.192·27-s − 1.88·29-s − 1.93·31-s − 0.809·33-s − 0.206·35-s + 1.16·37-s − 0.160·39-s + 0.848·41-s + 0.281·43-s − 0.217·45-s − 1.81·47-s − 0.899·49-s − 0.607·51-s + 1.20·53-s + 0.914·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 7.29T + 125T^{2} \) |
| 7 | \( 1 - 5.87T + 343T^{2} \) |
| 11 | \( 1 + 51.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 73.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 294.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 334.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 79.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 584.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 465.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 548.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 384.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 307.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 844.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 30.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 19.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 513.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 787.T + 9.12e5T^{2} \) |
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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
−10.95022448435788984664116694516, −9.734403784483738748995039249710, −8.842982570830159360008184520910, −7.72187164675514945245562756759, −7.34869108573832431349859057970, −5.64994998988630759394864214797, −4.55562015057282169677214473534, −3.33770158113213474685252645889, −2.06605078619900576989463972130, 0,
2.06605078619900576989463972130, 3.33770158113213474685252645889, 4.55562015057282169677214473534, 5.64994998988630759394864214797, 7.34869108573832431349859057970, 7.72187164675514945245562756759, 8.842982570830159360008184520910, 9.734403784483738748995039249710, 10.95022448435788984664116694516
