| L(s) = 1 | + 5·3-s − 6·5-s + 8·7-s − 2·9-s − 34·11-s − 57·13-s − 30·15-s − 80·17-s + 70·19-s + 40·21-s − 23·23-s − 89·25-s − 145·27-s + 245·29-s − 103·31-s − 170·33-s − 48·35-s − 298·37-s − 285·39-s + 95·41-s − 88·43-s + 12·45-s + 357·47-s − 279·49-s − 400·51-s − 414·53-s + 204·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.536·5-s + 0.431·7-s − 0.0740·9-s − 0.931·11-s − 1.21·13-s − 0.516·15-s − 1.14·17-s + 0.845·19-s + 0.415·21-s − 0.208·23-s − 0.711·25-s − 1.03·27-s + 1.56·29-s − 0.596·31-s − 0.896·33-s − 0.231·35-s − 1.32·37-s − 1.17·39-s + 0.361·41-s − 0.312·43-s + 0.0397·45-s + 1.10·47-s − 0.813·49-s − 1.09·51-s − 1.07·53-s + 0.500·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 34 T + p^{3} T^{2} \) |
| 13 | \( 1 + 57 T + p^{3} T^{2} \) |
| 17 | \( 1 + 80 T + p^{3} T^{2} \) |
| 19 | \( 1 - 70 T + p^{3} T^{2} \) |
| 29 | \( 1 - 245 T + p^{3} T^{2} \) |
| 31 | \( 1 + 103 T + p^{3} T^{2} \) |
| 37 | \( 1 + 298 T + p^{3} T^{2} \) |
| 41 | \( 1 - 95 T + p^{3} T^{2} \) |
| 43 | \( 1 + 88 T + p^{3} T^{2} \) |
| 47 | \( 1 - 357 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 408 T + p^{3} T^{2} \) |
| 61 | \( 1 - 822 T + p^{3} T^{2} \) |
| 67 | \( 1 + 926 T + p^{3} T^{2} \) |
| 71 | \( 1 + 335 T + p^{3} T^{2} \) |
| 73 | \( 1 + 899 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1322 T + p^{3} T^{2} \) |
| 83 | \( 1 - 36 T + p^{3} T^{2} \) |
| 89 | \( 1 + 460 T + p^{3} T^{2} \) |
| 97 | \( 1 + 964 T + p^{3} T^{2} \) |
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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
−10.45554460578583978425987817189, −9.490167238945498125657153099481, −8.509690026073376513068938275410, −7.85020120186757127913605052726, −7.04075806265867089474790267963, −5.43514373906710019938384034997, −4.41944755671317364201263474395, −3.09517395770447710001763738967, −2.14872934097866846527660044207, 0,
2.14872934097866846527660044207, 3.09517395770447710001763738967, 4.41944755671317364201263474395, 5.43514373906710019938384034997, 7.04075806265867089474790267963, 7.85020120186757127913605052726, 8.509690026073376513068938275410, 9.490167238945498125657153099481, 10.45554460578583978425987817189
