L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s − 2·11-s + 4·13-s + 2·14-s + 16-s + 2·17-s − 19-s − 20-s + 2·22-s − 4·23-s + 25-s − 4·26-s − 2·28-s − 8·31-s − 32-s − 2·34-s + 2·35-s + 8·37-s + 38-s + 40-s + 8·41-s − 6·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.223·20-s + 0.426·22-s − 0.834·23-s + 1/5·25-s − 0.784·26-s − 0.377·28-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.338·35-s + 1.31·37-s + 0.162·38-s + 0.158·40-s + 1.24·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8940592443\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8940592443\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.359140823907514452907770668441, −8.498251366635910760907557188074, −7.87174015309018254150043725183, −7.09731280092810077465364729019, −6.18445202309954802304210759452, −5.52386912324677339328619703846, −4.09538162677358768560618983547, −3.33653928678212644517465887249, −2.20525620187799007841675550757, −0.71105438939820272404677500706,
0.71105438939820272404677500706, 2.20525620187799007841675550757, 3.33653928678212644517465887249, 4.09538162677358768560618983547, 5.52386912324677339328619703846, 6.18445202309954802304210759452, 7.09731280092810077465364729019, 7.87174015309018254150043725183, 8.498251366635910760907557188074, 9.359140823907514452907770668441