| L(s) = 1 | − 0.0830·2-s + 2.10·3-s − 1.99·4-s + 5-s − 0.175·6-s + 7-s + 0.331·8-s + 1.44·9-s − 0.0830·10-s − 2.34·11-s − 4.20·12-s + 0.0413·13-s − 0.0830·14-s + 2.10·15-s + 3.95·16-s + 1.41·17-s − 0.120·18-s − 3.69·19-s − 1.99·20-s + 2.10·21-s + 0.194·22-s + 3.47·23-s + 0.699·24-s + 25-s − 0.00343·26-s − 3.27·27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | − 0.0587·2-s + 1.21·3-s − 0.996·4-s + 0.447·5-s − 0.0715·6-s + 0.377·7-s + 0.117·8-s + 0.481·9-s − 0.0262·10-s − 0.705·11-s − 1.21·12-s + 0.0114·13-s − 0.0222·14-s + 0.544·15-s + 0.989·16-s + 0.342·17-s − 0.0283·18-s − 0.847·19-s − 0.445·20-s + 0.460·21-s + 0.0414·22-s + 0.724·23-s + 0.142·24-s + 0.200·25-s − 0.000673·26-s − 0.630·27-s − 0.376·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0830T + 2T^{2} \) |
| 3 | \( 1 - 2.10T + 3T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 - 0.0413T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 3.69T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 + 0.644T + 29T^{2} \) |
| 31 | \( 1 + 9.97T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 - 6.22T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 - 2.91T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 3.76T + 67T^{2} \) |
| 71 | \( 1 + 7.28T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 8.13T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
| 97 | \( 1 + 0.367T + 97T^{2} \) |
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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
−7.74013223089267149941356870054, −7.07155652301842496031377860054, −5.91484695813625875232133173644, −5.32516012797212469105955988688, −4.62147858307847729794044149907, −3.78651173766179272990762789036, −3.14388006202774793518807683324, −2.27780303044565207131957413334, −1.44671805004439005588319521819, 0,
1.44671805004439005588319521819, 2.27780303044565207131957413334, 3.14388006202774793518807683324, 3.78651173766179272990762789036, 4.62147858307847729794044149907, 5.32516012797212469105955988688, 5.91484695813625875232133173644, 7.07155652301842496031377860054, 7.74013223089267149941356870054
