| L(s) = 1 | − 2-s − 2.93·3-s + 4-s − 0.199·5-s + 2.93·6-s − 7-s − 8-s + 5.59·9-s + 0.199·10-s − 4.44·11-s − 2.93·12-s − 5.18·13-s + 14-s + 0.585·15-s + 16-s + 1.80·17-s − 5.59·18-s + 8.15·19-s − 0.199·20-s + 2.93·21-s + 4.44·22-s + 2.93·24-s − 4.96·25-s + 5.18·26-s − 7.61·27-s − 28-s − 6.19·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.69·3-s + 0.5·4-s − 0.0893·5-s + 1.19·6-s − 0.377·7-s − 0.353·8-s + 1.86·9-s + 0.0631·10-s − 1.34·11-s − 0.846·12-s − 1.43·13-s + 0.267·14-s + 0.151·15-s + 0.250·16-s + 0.436·17-s − 1.31·18-s + 1.87·19-s − 0.0446·20-s + 0.639·21-s + 0.948·22-s + 0.598·24-s − 0.992·25-s + 1.01·26-s − 1.46·27-s − 0.188·28-s − 1.15·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2.93T + 3T^{2} \) |
| 5 | \( 1 + 0.199T + 5T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 5.18T + 13T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 - 8.15T + 19T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 + 3.03T + 43T^{2} \) |
| 47 | \( 1 - 6.58T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 - 9.19T + 59T^{2} \) |
| 61 | \( 1 + 4.00T + 61T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 - 0.0140T + 71T^{2} \) |
| 73 | \( 1 + 4.91T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 0.689T + 97T^{2} \) |
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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
−7.45287631524294093393154996985, −7.11103302977766328691227717334, −5.98094934821140175019636087232, −5.48402215439293440535806039440, −5.15383684779416206225483614193, −4.07874737251560754458104311860, −2.98674611557305848902358504381, −2.03816846603451166712811732785, −0.78509138224854435201693396120, 0,
0.78509138224854435201693396120, 2.03816846603451166712811732785, 2.98674611557305848902358504381, 4.07874737251560754458104311860, 5.15383684779416206225483614193, 5.48402215439293440535806039440, 5.98094934821140175019636087232, 7.11103302977766328691227717334, 7.45287631524294093393154996985
