| L(s) = 1 | − 2-s − 0.968·3-s + 4-s − 0.0731·5-s + 0.968·6-s − 0.501·7-s − 8-s − 2.06·9-s + 0.0731·10-s − 1.89·11-s − 0.968·12-s + 5.36·13-s + 0.501·14-s + 0.0708·15-s + 16-s + 0.196·17-s + 2.06·18-s − 5.86·19-s − 0.0731·20-s + 0.485·21-s + 1.89·22-s + 23-s + 0.968·24-s − 4.99·25-s − 5.36·26-s + 4.90·27-s − 0.501·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.559·3-s + 0.5·4-s − 0.0327·5-s + 0.395·6-s − 0.189·7-s − 0.353·8-s − 0.687·9-s + 0.0231·10-s − 0.570·11-s − 0.279·12-s + 1.48·13-s + 0.133·14-s + 0.0183·15-s + 0.250·16-s + 0.0476·17-s + 0.485·18-s − 1.34·19-s − 0.0163·20-s + 0.105·21-s + 0.403·22-s + 0.208·23-s + 0.197·24-s − 0.998·25-s − 1.05·26-s + 0.943·27-s − 0.0947·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6493142091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6493142091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 0.968T + 3T^{2} \) |
| 5 | \( 1 + 0.0731T + 5T^{2} \) |
| 7 | \( 1 + 0.501T + 7T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 - 0.196T + 17T^{2} \) |
| 19 | \( 1 + 5.86T + 19T^{2} \) |
| 29 | \( 1 + 7.01T + 29T^{2} \) |
| 31 | \( 1 + 0.0479T + 31T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 1.74T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 - 2.64T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 0.148T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 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
−8.032526185298920778527231591512, −7.60413601753507775555850125190, −6.44012600396764122472967108535, −6.10478476502740242776504294012, −5.52014987416887449042530028817, −4.42454047017552027308486875656, −3.57155587491749739811616511006, −2.65351719349469180257873913164, −1.69076728146837005540189238050, −0.47807127549924087450046046278,
0.47807127549924087450046046278, 1.69076728146837005540189238050, 2.65351719349469180257873913164, 3.57155587491749739811616511006, 4.42454047017552027308486875656, 5.52014987416887449042530028817, 6.10478476502740242776504294012, 6.44012600396764122472967108535, 7.60413601753507775555850125190, 8.032526185298920778527231591512
