| L(s) = 1 | + 2-s + 3-s + 4-s + 4.22·5-s + 6-s + 0.162·7-s + 8-s + 9-s + 4.22·10-s + 4.28·11-s + 12-s − 1.72·13-s + 0.162·14-s + 4.22·15-s + 16-s − 0.372·17-s + 18-s + 19-s + 4.22·20-s + 0.162·21-s + 4.28·22-s − 3.58·23-s + 24-s + 12.8·25-s − 1.72·26-s + 27-s + 0.162·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.89·5-s + 0.408·6-s + 0.0614·7-s + 0.353·8-s + 0.333·9-s + 1.33·10-s + 1.29·11-s + 0.288·12-s − 0.479·13-s + 0.0434·14-s + 1.09·15-s + 0.250·16-s − 0.0903·17-s + 0.235·18-s + 0.229·19-s + 0.945·20-s + 0.0354·21-s + 0.914·22-s − 0.747·23-s + 0.204·24-s + 2.57·25-s − 0.339·26-s + 0.192·27-s + 0.0307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.435889091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.435889091\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 4.22T + 5T^{2} \) |
| 7 | \( 1 - 0.162T + 7T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 - 9.28T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 - 1.48T + 37T^{2} \) |
| 41 | \( 1 + 2.88T + 41T^{2} \) |
| 43 | \( 1 + 7.90T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 7.40T + 79T^{2} \) |
| 83 | \( 1 + 8.64T + 83T^{2} \) |
| 89 | \( 1 - 3.54T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−8.132347594780048105800592654330, −7.08366969855441484243770934891, −6.41763163527411931874322952733, −6.12662632006544320258575860250, −5.08580567538728187916513050228, −4.62292582040194951792759550365, −3.51260073709961230059081830105, −2.78369098600431010487963968790, −1.90057387207596089024532403107, −1.37409726128445129593198700170,
1.37409726128445129593198700170, 1.90057387207596089024532403107, 2.78369098600431010487963968790, 3.51260073709961230059081830105, 4.62292582040194951792759550365, 5.08580567538728187916513050228, 6.12662632006544320258575860250, 6.41763163527411931874322952733, 7.08366969855441484243770934891, 8.132347594780048105800592654330
