| L(s) = 1 | + 3.30·3-s − 5-s + 0.302·7-s + 7.90·9-s − 5.30·11-s + 0.302·13-s − 3.30·15-s − 3.90·17-s − 4.90·19-s + 1.00·21-s + 23-s + 25-s + 16.2·27-s − 4.60·29-s − 2.90·31-s − 17.5·33-s − 0.302·35-s − 8·37-s + 1.00·39-s − 9.90·41-s + 5.21·43-s − 7.90·45-s − 4.60·47-s − 6.90·49-s − 12.9·51-s − 3.21·53-s + 5.30·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.447·5-s + 0.114·7-s + 2.63·9-s − 1.59·11-s + 0.0839·13-s − 0.852·15-s − 0.947·17-s − 1.12·19-s + 0.218·21-s + 0.208·23-s + 0.200·25-s + 3.11·27-s − 0.855·29-s − 0.522·31-s − 3.04·33-s − 0.0511·35-s − 1.31·37-s + 0.160·39-s − 1.54·41-s + 0.794·43-s − 1.17·45-s − 0.671·47-s − 0.986·49-s − 1.80·51-s − 0.441·53-s + 0.715·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 - 0.302T + 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 - 0.302T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 + 2.90T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 6.51T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2.69T + 97T^{2} \) |
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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
−7.70527848534441475907281428470, −7.17528990358384798006758400352, −6.44734435981539079491873721704, −5.14764876800815943556699820203, −4.57029156175093518634925353867, −3.70772645373228457850220670383, −3.14756312078567539697371469705, −2.28709854340917422654663263249, −1.77869044322773797163803002012, 0,
1.77869044322773797163803002012, 2.28709854340917422654663263249, 3.14756312078567539697371469705, 3.70772645373228457850220670383, 4.57029156175093518634925353867, 5.14764876800815943556699820203, 6.44734435981539079491873721704, 7.17528990358384798006758400352, 7.70527848534441475907281428470
