| L(s) = 1 | − 3-s − 2·4-s − 5-s + 9-s − 2·11-s + 2·12-s + 13-s + 15-s + 4·16-s + 2·17-s + 5·19-s + 2·20-s + 23-s − 4·25-s − 27-s − 3·29-s + 31-s + 2·33-s − 2·36-s + 4·37-s − 39-s + 6·41-s − 43-s + 4·44-s − 45-s − 3·47-s − 4·48-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.577·12-s + 0.277·13-s + 0.258·15-s + 16-s + 0.485·17-s + 1.14·19-s + 0.447·20-s + 0.208·23-s − 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.179·31-s + 0.348·33-s − 1/3·36-s + 0.657·37-s − 0.160·39-s + 0.937·41-s − 0.152·43-s + 0.603·44-s − 0.149·45-s − 0.437·47-s − 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.906499724308941141962814398979, −7.77937202450307017756452622506, −7.62412684322283631414642839520, −6.22280635515153117547178435548, −5.50076674524025886548347180073, −4.78441256465789525523044549972, −3.93310228709688602926654473178, −3.02984768018332177307923477692, −1.27641806466759505307706490567, 0,
1.27641806466759505307706490567, 3.02984768018332177307923477692, 3.93310228709688602926654473178, 4.78441256465789525523044549972, 5.50076674524025886548347180073, 6.22280635515153117547178435548, 7.62412684322283631414642839520, 7.77937202450307017756452622506, 8.906499724308941141962814398979
