| L(s) = 1 | + 2-s − 1.61·3-s + 4-s − 1.61·5-s − 1.61·6-s + 3.85·7-s + 8-s − 0.381·9-s − 1.61·10-s − 1.61·12-s − 0.618·13-s + 3.85·14-s + 2.61·15-s + 16-s − 3.23·17-s − 0.381·18-s + 19-s − 1.61·20-s − 6.23·21-s − 2·23-s − 1.61·24-s − 2.38·25-s − 0.618·26-s + 5.47·27-s + 3.85·28-s + 0.854·29-s + 2.61·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.934·3-s + 0.5·4-s − 0.723·5-s − 0.660·6-s + 1.45·7-s + 0.353·8-s − 0.127·9-s − 0.511·10-s − 0.467·12-s − 0.171·13-s + 1.03·14-s + 0.675·15-s + 0.250·16-s − 0.784·17-s − 0.0900·18-s + 0.229·19-s − 0.361·20-s − 1.36·21-s − 0.417·23-s − 0.330·24-s − 0.476·25-s − 0.121·26-s + 1.05·27-s + 0.728·28-s + 0.158·29-s + 0.477·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 0.854T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 - 1.23T + 37T^{2} \) |
| 41 | \( 1 - 0.618T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 0.763T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 2.38T + 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.079669302764907765894245251210, −7.06641467689381830597031919905, −6.43653090598679040491518179036, −5.51707681286640136353231277711, −5.01350273666736267240843715141, −4.40756861604253388728898886618, −3.61046177234654650859244430183, −2.42482992612592506085229127537, −1.42460999369936152160007923338, 0,
1.42460999369936152160007923338, 2.42482992612592506085229127537, 3.61046177234654650859244430183, 4.40756861604253388728898886618, 5.01350273666736267240843715141, 5.51707681286640136353231277711, 6.43653090598679040491518179036, 7.06641467689381830597031919905, 8.079669302764907765894245251210
