| L(s) = 1 | − 1.62·3-s − 4.74·7-s − 0.367·9-s + 4.48·11-s − 0.843·13-s − 5.52·17-s + 19-s + 7.69·21-s + 0.779·23-s + 5.46·27-s − 10.6·29-s − 8.65·31-s − 7.26·33-s − 1.62·37-s + 1.36·39-s + 4.73·41-s + 9.67·43-s − 3.18·47-s + 15.5·49-s + 8.96·51-s + 6.17·53-s − 1.62·57-s + 11.6·59-s + 6.48·61-s + 1.74·63-s + 14.8·67-s − 1.26·69-s + ⋯ |
| L(s) = 1 | − 0.936·3-s − 1.79·7-s − 0.122·9-s + 1.35·11-s − 0.233·13-s − 1.33·17-s + 0.229·19-s + 1.67·21-s + 0.162·23-s + 1.05·27-s − 1.97·29-s − 1.55·31-s − 1.26·33-s − 0.266·37-s + 0.219·39-s + 0.739·41-s + 1.47·43-s − 0.464·47-s + 2.21·49-s + 1.25·51-s + 0.848·53-s − 0.214·57-s + 1.52·59-s + 0.829·61-s + 0.219·63-s + 1.81·67-s − 0.152·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6378818216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6378818216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.62T + 3T^{2} \) |
| 7 | \( 1 + 4.74T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 + 0.843T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 23 | \( 1 - 0.779T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 + 1.62T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 - 9.67T + 43T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 - 6.17T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 0.303T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 0.779T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 6.17T + 97T^{2} \) |
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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
−9.258511744535299582162002244154, −8.779309504631923296370568401122, −7.18169834560311729195130255998, −6.82319593493255798845869425117, −6.00569609876773148642158085481, −5.48930011817670878195103570678, −4.14016570498397586212028156149, −3.50519090073751939483112350586, −2.24622620408940308031995104146, −0.53756157683053079461003563806,
0.53756157683053079461003563806, 2.24622620408940308031995104146, 3.50519090073751939483112350586, 4.14016570498397586212028156149, 5.48930011817670878195103570678, 6.00569609876773148642158085481, 6.82319593493255798845869425117, 7.18169834560311729195130255998, 8.779309504631923296370568401122, 9.258511744535299582162002244154
