L(s) = 1 | − 2-s + 4-s + 2·5-s + 7-s − 8-s − 2·10-s − 6·11-s − 4·13-s − 14-s + 16-s + 2·17-s + 4·19-s + 2·20-s + 6·22-s − 23-s − 25-s + 4·26-s + 28-s + 10·29-s − 8·31-s − 32-s − 2·34-s + 2·35-s − 8·37-s − 4·38-s − 2·40-s + 2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s − 1.80·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s + 1.27·22-s − 0.208·23-s − 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.85·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.338·35-s − 1.31·37-s − 0.648·38-s − 0.316·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.308187831212357090485335492252, −7.69593888538343968821370678965, −7.15288228516136701752701113567, −6.05406748629782483135723000821, −5.31624768297914470660739707361, −4.81118102361524102592547966816, −3.16640640810488659483297309312, −2.45977329276086919056996653075, −1.54796165697611857024905292769, 0,
1.54796165697611857024905292769, 2.45977329276086919056996653075, 3.16640640810488659483297309312, 4.81118102361524102592547966816, 5.31624768297914470660739707361, 6.05406748629782483135723000821, 7.15288228516136701752701113567, 7.69593888538343968821370678965, 8.308187831212357090485335492252