| L(s) = 1 | − 2·3-s − 2·4-s − 5-s − 2·7-s + 9-s − 2·11-s + 4·12-s + 13-s + 2·15-s + 4·16-s + 3·17-s − 5·19-s + 2·20-s + 4·21-s + 23-s − 4·25-s + 4·27-s + 4·28-s − 4·29-s − 9·31-s + 4·33-s + 2·35-s − 2·36-s − 2·37-s − 2·39-s + 8·41-s − 8·43-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 1.15·12-s + 0.277·13-s + 0.516·15-s + 16-s + 0.727·17-s − 1.14·19-s + 0.447·20-s + 0.872·21-s + 0.208·23-s − 4/5·25-s + 0.769·27-s + 0.755·28-s − 0.742·29-s − 1.61·31-s + 0.696·33-s + 0.338·35-s − 1/3·36-s − 0.328·37-s − 0.320·39-s + 1.24·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 101 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.00508217019688623012513185043, −12.46381898373739464831556120111, −11.18987953654912073880974227563, −10.24202176919572853588664276585, −9.060317898736739853083259611177, −7.74187934910278185224762610896, −6.16055116457811093791815485098, −5.17330498735283860298501426044, −3.72933468340090913789145872057, 0,
3.72933468340090913789145872057, 5.17330498735283860298501426044, 6.16055116457811093791815485098, 7.74187934910278185224762610896, 9.060317898736739853083259611177, 10.24202176919572853588664276585, 11.18987953654912073880974227563, 12.46381898373739464831556120111, 13.00508217019688623012513185043
