L(s) = 1 | + 1.41·3-s − 4.24·5-s − 0.999·9-s + 11-s − 6·15-s + 5.65·17-s − 6·23-s + 12.9·25-s − 5.65·27-s + 2·29-s − 1.41·31-s + 1.41·33-s − 10·37-s + 11.3·41-s + 8·43-s + 4.24·45-s + 4.24·47-s + 8.00·51-s + 8·53-s − 4.24·55-s + 1.41·59-s + 2.82·61-s − 2·67-s − 8.48·69-s + 2·71-s − 8.48·73-s + 18.3·75-s + ⋯ |
L(s) = 1 | + 0.816·3-s − 1.89·5-s − 0.333·9-s + 0.301·11-s − 1.54·15-s + 1.37·17-s − 1.25·23-s + 2.59·25-s − 1.08·27-s + 0.371·29-s − 0.254·31-s + 0.246·33-s − 1.64·37-s + 1.76·41-s + 1.21·43-s + 0.632·45-s + 0.618·47-s + 1.12·51-s + 1.09·53-s − 0.572·55-s + 0.184·59-s + 0.362·61-s − 0.244·67-s − 1.02·69-s + 0.237·71-s − 0.993·73-s + 2.12·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 4.24T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.55919042915053272732313812014, −7.15284625046120612477138464860, −6.02724626711405804352889828388, −5.33208810314140874354758069471, −4.20253687837471580472924293538, −3.88829186311956508559274793268, −3.19758987670788427113794821729, −2.48213639822033695202223575691, −1.12730952737508724650815682186, 0,
1.12730952737508724650815682186, 2.48213639822033695202223575691, 3.19758987670788427113794821729, 3.88829186311956508559274793268, 4.20253687837471580472924293538, 5.33208810314140874354758069471, 6.02724626711405804352889828388, 7.15284625046120612477138464860, 7.55919042915053272732313812014