| L(s) = 1 | − 0.381·2-s − 0.381·3-s − 1.85·4-s + 0.145·6-s + 4.23·7-s + 1.47·8-s − 2.85·9-s − 2.38·11-s + 0.708·12-s − 5·13-s − 1.61·14-s + 3.14·16-s + 6·17-s + 1.09·18-s − 1.61·21-s + 0.909·22-s − 0.618·23-s − 0.562·24-s + 1.90·26-s + 2.23·27-s − 7.85·28-s + 4.85·29-s + 10.8·31-s − 4.14·32-s + 0.909·33-s − 2.29·34-s + 5.29·36-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.220·3-s − 0.927·4-s + 0.0595·6-s + 1.60·7-s + 0.520·8-s − 0.951·9-s − 0.718·11-s + 0.204·12-s − 1.38·13-s − 0.432·14-s + 0.786·16-s + 1.45·17-s + 0.256·18-s − 0.353·21-s + 0.193·22-s − 0.128·23-s − 0.114·24-s + 0.374·26-s + 0.430·27-s − 1.48·28-s + 0.901·29-s + 1.94·31-s − 0.732·32-s + 0.158·33-s − 0.393·34-s + 0.881·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122669652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122669652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 23 | \( 1 + 0.618T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 3.85T + 43T^{2} \) |
| 47 | \( 1 - 5.76T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 - 5.09T + 97T^{2} \) |
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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
−7.972848467038277614207697710298, −7.38672295160848978584028566035, −6.25426324543796476160813314911, −5.34562781775155113221910655341, −4.94197422585791770343915639731, −4.65737892637451429163093685866, −3.40780357621589447909614807523, −2.61595613796707192788643105114, −1.56272071381233982663400079465, −0.56479220947606063370992410294,
0.56479220947606063370992410294, 1.56272071381233982663400079465, 2.61595613796707192788643105114, 3.40780357621589447909614807523, 4.65737892637451429163093685866, 4.94197422585791770343915639731, 5.34562781775155113221910655341, 6.25426324543796476160813314911, 7.38672295160848978584028566035, 7.972848467038277614207697710298
