| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s − 4.24·13-s + 16-s − 2.82·17-s + 4.24·19-s − 22-s − 6·23-s − 5·25-s + 4.24·26-s + 4·29-s + 7.07·31-s − 32-s + 2.82·34-s + 2·37-s − 4.24·38-s + 2.82·41-s + 10·43-s + 44-s + 6·46-s − 12.7·47-s + 5·50-s − 4.24·52-s − 2·53-s − 4·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.353·8-s + 0.301·11-s − 1.17·13-s + 0.250·16-s − 0.685·17-s + 0.973·19-s − 0.213·22-s − 1.25·23-s − 25-s + 0.832·26-s + 0.742·29-s + 1.27·31-s − 0.176·32-s + 0.485·34-s + 0.328·37-s − 0.688·38-s + 0.441·41-s + 1.52·43-s + 0.150·44-s + 0.884·46-s − 1.85·47-s + 0.707·50-s − 0.588·52-s − 0.274·53-s − 0.525·58-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 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−7.49151797748353724339772904966, −6.73434738171616199134482069220, −6.15284543364006056548718299211, −5.35891266185092378158829619906, −4.53764696532617191512962115334, −3.81771704539733165727045683571, −2.73373241935686791236895631579, −2.19371925871591313739375559171, −1.10508380038814680699142894786, 0,
1.10508380038814680699142894786, 2.19371925871591313739375559171, 2.73373241935686791236895631579, 3.81771704539733165727045683571, 4.53764696532617191512962115334, 5.35891266185092378158829619906, 6.15284543364006056548718299211, 6.73434738171616199134482069220, 7.49151797748353724339772904966
