L(s) = 1 | − 2·2-s + 4·4-s − 3·5-s + 7·7-s − 8·8-s + 6·10-s + 11·11-s + 41·13-s − 14·14-s + 16·16-s − 6·17-s − 43·19-s − 12·20-s − 22·22-s − 120·23-s − 116·25-s − 82·26-s + 28·28-s − 111·29-s + 266·31-s − 32·32-s + 12·34-s − 21·35-s − 79·37-s + 86·38-s + 24·40-s − 216·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.268·5-s + 0.377·7-s − 0.353·8-s + 0.189·10-s + 0.301·11-s + 0.874·13-s − 0.267·14-s + 1/4·16-s − 0.0856·17-s − 0.519·19-s − 0.134·20-s − 0.213·22-s − 1.08·23-s − 0.927·25-s − 0.618·26-s + 0.188·28-s − 0.710·29-s + 1.54·31-s − 0.176·32-s + 0.0605·34-s − 0.101·35-s − 0.351·37-s + 0.367·38-s + 0.0948·40-s − 0.822·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
good | 5 | \( 1 + 3 T + p^{3} T^{2} \) |
| 13 | \( 1 - 41 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 + 43 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 111 T + p^{3} T^{2} \) |
| 31 | \( 1 - 266 T + p^{3} T^{2} \) |
| 37 | \( 1 + 79 T + p^{3} T^{2} \) |
| 41 | \( 1 + 216 T + p^{3} T^{2} \) |
| 43 | \( 1 - 284 T + p^{3} T^{2} \) |
| 47 | \( 1 + 213 T + p^{3} T^{2} \) |
| 53 | \( 1 - 216 T + p^{3} T^{2} \) |
| 59 | \( 1 + 393 T + p^{3} T^{2} \) |
| 61 | \( 1 - 350 T + p^{3} T^{2} \) |
| 67 | \( 1 - 821 T + p^{3} T^{2} \) |
| 71 | \( 1 - 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 865 T + p^{3} T^{2} \) |
| 79 | \( 1 + 484 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1158 T + p^{3} T^{2} \) |
| 89 | \( 1 + 330 T + p^{3} T^{2} \) |
| 97 | \( 1 - 980 T + p^{3} T^{2} \) |
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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
−8.592581425148084401920757488349, −8.222552021114660242433348568264, −7.33183622986043499678808230232, −6.39964827659824922233042458134, −5.70222973852395637493389827930, −4.40168393276223592932355444060, −3.58788784566443930872412043770, −2.27163114157772339316552266883, −1.28147247346995776795387367784, 0,
1.28147247346995776795387367784, 2.27163114157772339316552266883, 3.58788784566443930872412043770, 4.40168393276223592932355444060, 5.70222973852395637493389827930, 6.39964827659824922233042458134, 7.33183622986043499678808230232, 8.222552021114660242433348568264, 8.592581425148084401920757488349