L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 3·9-s − 10-s + 6·13-s + 14-s + 16-s − 6·17-s − 3·18-s − 6·19-s − 20-s + 8·23-s + 25-s + 6·26-s + 28-s + 6·29-s − 8·31-s + 32-s − 6·34-s − 35-s − 3·36-s − 4·37-s − 6·38-s − 40-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 1.37·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 0.169·35-s − 1/2·36-s − 0.657·37-s − 0.973·38-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.26957233392619331027865419665, −6.49620425011047594820310134681, −6.25394887123111955610183625395, −5.14178817519770172544023602709, −4.76152979125461987655533869998, −3.78108926342181475102700978120, −3.29932396260114917305957664119, −2.33950756845724146791082842596, −1.42475620338215146764176473083, 0,
1.42475620338215146764176473083, 2.33950756845724146791082842596, 3.29932396260114917305957664119, 3.78108926342181475102700978120, 4.76152979125461987655533869998, 5.14178817519770172544023602709, 6.25394887123111955610183625395, 6.49620425011047594820310134681, 7.26957233392619331027865419665