L(s) = 1 | + 2.23·2-s + 3.00·4-s + 2·5-s − 7-s + 2.23·8-s + 4.47·10-s − 3.23·13-s − 2.23·14-s − 0.999·16-s − 3.23·17-s − 6.47·19-s + 6.00·20-s − 2.47·23-s − 25-s − 7.23·26-s − 3.00·28-s + 8.47·29-s − 2.76·31-s − 6.70·32-s − 7.23·34-s − 2·35-s − 8.47·37-s − 14.4·38-s + 4.47·40-s − 11.2·41-s − 8·43-s − 5.52·46-s + ⋯ |
L(s) = 1 | + 1.58·2-s + 1.50·4-s + 0.894·5-s − 0.377·7-s + 0.790·8-s + 1.41·10-s − 0.897·13-s − 0.597·14-s − 0.249·16-s − 0.784·17-s − 1.48·19-s + 1.34·20-s − 0.515·23-s − 0.200·25-s − 1.41·26-s − 0.566·28-s + 1.57·29-s − 0.496·31-s − 1.18·32-s − 1.24·34-s − 0.338·35-s − 1.39·37-s − 2.34·38-s + 0.707·40-s − 1.75·41-s − 1.21·43-s − 0.815·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 0.763T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−6.92485576817166123002217388980, −6.67615697970454416412119202383, −6.10091394869788754672873127488, −5.20127743241274905586898795999, −4.86843857665934763990730064093, −3.98621323110010483607957956695, −3.28395844464187692191426110187, −2.27704977660579191787807640991, −1.97058496082294878063569811738, 0,
1.97058496082294878063569811738, 2.27704977660579191787807640991, 3.28395844464187692191426110187, 3.98621323110010483607957956695, 4.86843857665934763990730064093, 5.20127743241274905586898795999, 6.10091394869788754672873127488, 6.67615697970454416412119202383, 6.92485576817166123002217388980