| L(s) = 1 | + 1.13·2-s − 1.94·3-s + 0.290·4-s + 5-s − 2.20·6-s − 1.49·7-s − 0.805·8-s + 2.77·9-s + 1.13·10-s + 0.241·11-s − 0.564·12-s + 13-s − 1.70·14-s − 1.94·15-s − 1.20·16-s − 0.709·17-s + 3.14·18-s + 0.290·20-s + 2.90·21-s + 0.273·22-s + 1.77·23-s + 1.56·24-s + 25-s + 1.13·26-s − 3.43·27-s − 0.435·28-s − 2.20·30-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 1.94·3-s + 0.290·4-s + 5-s − 2.20·6-s − 1.49·7-s − 0.805·8-s + 2.77·9-s + 1.13·10-s + 0.241·11-s − 0.564·12-s + 13-s − 1.70·14-s − 1.94·15-s − 1.20·16-s − 0.709·17-s + 3.14·18-s + 0.290·20-s + 2.90·21-s + 0.273·22-s + 1.77·23-s + 1.56·24-s + 25-s + 1.13·26-s − 3.43·27-s − 0.435·28-s − 2.20·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.053836155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.053836155\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.13T + T^{2} \) |
| 3 | \( 1 + 1.94T + T^{2} \) |
| 7 | \( 1 + 1.49T + T^{2} \) |
| 11 | \( 1 - 0.241T + T^{2} \) |
| 17 | \( 1 + 0.709T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.77T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.13T + T^{2} \) |
| 47 | \( 1 - 1.77T + T^{2} \) |
| 53 | \( 1 - 0.241T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.241T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.49T + T^{2} \) |
| 97 | \( 1 + 1.94T + T^{2} \) |
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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
−9.452769635684117173465800360952, −8.962377376857449546465885940728, −6.92789041691993445342646562261, −6.64103858910678542765177074989, −5.94232608495647466110936982819, −5.56249394481462576086985572357, −4.65286380893871379178749336942, −3.88149702926570741084698361077, −2.72707226269643274988174939378, −0.967595316357987748728055239231,
0.967595316357987748728055239231, 2.72707226269643274988174939378, 3.88149702926570741084698361077, 4.65286380893871379178749336942, 5.56249394481462576086985572357, 5.94232608495647466110936982819, 6.64103858910678542765177074989, 6.92789041691993445342646562261, 8.962377376857449546465885940728, 9.452769635684117173465800360952
