| L(s) = 1 | − 2.08·2-s + 3-s + 2.35·4-s − 2.08·6-s + 4.08·7-s − 0.734·8-s + 9-s − 3.43·11-s + 2.35·12-s + 13-s − 8.52·14-s − 3.17·16-s + 7.17·17-s − 2.08·18-s + 7.52·19-s + 4.08·21-s + 7.17·22-s − 2.82·23-s − 0.734·24-s − 2.08·26-s + 27-s + 9.61·28-s − 5.70·29-s − 5.61·31-s + 8.08·32-s − 3.43·33-s − 14.9·34-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 0.577·3-s + 1.17·4-s − 0.851·6-s + 1.54·7-s − 0.259·8-s + 0.333·9-s − 1.03·11-s + 0.678·12-s + 0.277·13-s − 2.27·14-s − 0.793·16-s + 1.73·17-s − 0.491·18-s + 1.72·19-s + 0.891·21-s + 1.52·22-s − 0.588·23-s − 0.149·24-s − 0.409·26-s + 0.192·27-s + 1.81·28-s − 1.05·29-s − 1.00·31-s + 1.42·32-s − 0.598·33-s − 2.56·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.146083354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.146083354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 17 | \( 1 - 7.17T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 5.61T + 31T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 - 5.52T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 7.49T + 47T^{2} \) |
| 53 | \( 1 + 2.52T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 + 7.87T + 61T^{2} \) |
| 67 | \( 1 + 6.20T + 67T^{2} \) |
| 71 | \( 1 + 0.475T + 71T^{2} \) |
| 73 | \( 1 - 9.35T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−9.787454457072611051281101665740, −9.230662024308340007174631495839, −8.105829596598201450163565126363, −7.82320184195774979371782732484, −7.38268734409679032935553148040, −5.67375423496163735253308109132, −4.85892325027071397100505806941, −3.40387342268018249934370676156, −2.05646123861250513916152863171, −1.11693003239139283000714767590,
1.11693003239139283000714767590, 2.05646123861250513916152863171, 3.40387342268018249934370676156, 4.85892325027071397100505806941, 5.67375423496163735253308109132, 7.38268734409679032935553148040, 7.82320184195774979371782732484, 8.105829596598201450163565126363, 9.230662024308340007174631495839, 9.787454457072611051281101665740
