L(s) = 1 | − 2-s + 4-s − 3.27·5-s + 0.271·7-s − 8-s + 3.27·10-s − 5.38·11-s + 3.97·13-s − 0.271·14-s + 16-s + 2.00·17-s + 4.41·19-s − 3.27·20-s + 5.38·22-s − 23-s + 5.71·25-s − 3.97·26-s + 0.271·28-s + 2.38·29-s − 8.86·31-s − 32-s − 2.00·34-s − 0.888·35-s − 5.59·37-s − 4.41·38-s + 3.27·40-s + 4.48·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.46·5-s + 0.102·7-s − 0.353·8-s + 1.03·10-s − 1.62·11-s + 1.10·13-s − 0.0725·14-s + 0.250·16-s + 0.485·17-s + 1.01·19-s − 0.732·20-s + 1.14·22-s − 0.208·23-s + 1.14·25-s − 0.779·26-s + 0.0512·28-s + 0.442·29-s − 1.59·31-s − 0.176·32-s − 0.343·34-s − 0.150·35-s − 0.919·37-s − 0.716·38-s + 0.517·40-s + 0.700·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 3.27T + 5T^{2} \) |
| 7 | \( 1 - 0.271T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 - 3.97T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 + 8.86T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 - 4.48T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.63T + 47T^{2} \) |
| 53 | \( 1 - 0.510T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 7.94T + 71T^{2} \) |
| 73 | \( 1 + 5.62T + 73T^{2} \) |
| 79 | \( 1 + 0.0359T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.206407437636442450727276686387, −7.42161802188371069368518824640, −7.21174080203943439886256974334, −5.83820755737859599388850485737, −5.27636891844194388802583522786, −4.10775490255661020292971554093, −3.41573626880027921078082001236, −2.54744952118320411596550032482, −1.11788813112118339904908130944, 0,
1.11788813112118339904908130944, 2.54744952118320411596550032482, 3.41573626880027921078082001236, 4.10775490255661020292971554093, 5.27636891844194388802583522786, 5.83820755737859599388850485737, 7.21174080203943439886256974334, 7.42161802188371069368518824640, 8.206407437636442450727276686387