| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 11-s + 5.41·13-s + 16-s − 2·17-s − 2.24·19-s − 2·20-s − 22-s + 4.82·23-s − 25-s − 5.41·26-s − 9.65·29-s + 6.24·31-s − 32-s + 2·34-s + 0.828·37-s + 2.24·38-s + 2·40-s − 11.6·41-s − 4.82·43-s + 44-s − 4.82·46-s + 7.89·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.50·13-s + 0.250·16-s − 0.485·17-s − 0.514·19-s − 0.447·20-s − 0.213·22-s + 1.00·23-s − 0.200·25-s − 1.06·26-s − 1.79·29-s + 1.12·31-s − 0.176·32-s + 0.342·34-s + 0.136·37-s + 0.363·38-s + 0.316·40-s − 1.82·41-s − 0.736·43-s + 0.150·44-s − 0.711·46-s + 1.15·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 13 | \( 1 - 5.41T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 9.65T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 0.828T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 7.89T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 6.82T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 5.07T + 83T^{2} \) |
| 89 | \( 1 + 3.75T + 89T^{2} \) |
| 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
−7.34631135086647411826343601970, −6.84398101536375019088910068645, −6.14100495040915870233285696474, −5.41473216242024660065866511989, −4.35313128952855843786545992903, −3.77095578713166586613405112466, −3.09015424304817645981087603647, −1.96894898700370322778528809264, −1.09552030927412792905760582792, 0,
1.09552030927412792905760582792, 1.96894898700370322778528809264, 3.09015424304817645981087603647, 3.77095578713166586613405112466, 4.35313128952855843786545992903, 5.41473216242024660065866511989, 6.14100495040915870233285696474, 6.84398101536375019088910068645, 7.34631135086647411826343601970
