| L(s) = 1 | + 2.02·2-s − 1.91·3-s + 2.08·4-s − 3.86·6-s − 7-s + 0.163·8-s + 0.663·9-s − 0.288·11-s − 3.98·12-s + 1.24·13-s − 2.02·14-s − 3.83·16-s + 2.73·17-s + 1.34·18-s − 1.21·19-s + 1.91·21-s − 0.583·22-s + 23-s − 0.313·24-s + 2.52·26-s + 4.47·27-s − 2.08·28-s + 8.65·29-s − 5.24·31-s − 8.06·32-s + 0.552·33-s + 5.52·34-s + ⋯ |
| L(s) = 1 | + 1.42·2-s − 1.10·3-s + 1.04·4-s − 1.57·6-s − 0.377·7-s + 0.0578·8-s + 0.221·9-s − 0.0871·11-s − 1.14·12-s + 0.346·13-s − 0.539·14-s − 0.957·16-s + 0.663·17-s + 0.316·18-s − 0.278·19-s + 0.417·21-s − 0.124·22-s + 0.208·23-s − 0.0639·24-s + 0.494·26-s + 0.860·27-s − 0.393·28-s + 1.60·29-s − 0.941·31-s − 1.42·32-s + 0.0962·33-s + 0.947·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.229563344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.229563344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 11 | \( 1 + 0.288T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 5.72T + 41T^{2} \) |
| 43 | \( 1 - 4.87T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 9.91T + 71T^{2} \) |
| 73 | \( 1 - 3.84T + 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 2.42T + 97T^{2} \) |
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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
−8.481641716958068707360649641480, −7.24012622481966877411381750566, −6.63537653143591663164811682585, −6.01434556425401444172525550402, −5.36706188314183148511015221597, −4.92603686912328816028898846898, −3.92996330006067492227549739202, −3.28224674539582691234203868985, −2.26984568610796584626902493164, −0.71194409535736708582045698988,
0.71194409535736708582045698988, 2.26984568610796584626902493164, 3.28224674539582691234203868985, 3.92996330006067492227549739202, 4.92603686912328816028898846898, 5.36706188314183148511015221597, 6.01434556425401444172525550402, 6.63537653143591663164811682585, 7.24012622481966877411381750566, 8.481641716958068707360649641480
