| L(s) = 1 | − 2-s + 2.73·3-s + 4-s − 2.73·6-s + 3·7-s − 8-s + 4.46·9-s + 3·11-s + 2.73·12-s − 3·14-s + 16-s − 2.19·17-s − 4.46·18-s + 6.46·19-s + 8.19·21-s − 3·22-s + 2.53·23-s − 2.73·24-s + 3.99·27-s + 3·28-s − 9.46·29-s + 1.26·31-s − 32-s + 8.19·33-s + 2.19·34-s + 4.46·36-s + 11.1·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s − 1.11·6-s + 1.13·7-s − 0.353·8-s + 1.48·9-s + 0.904·11-s + 0.788·12-s − 0.801·14-s + 0.250·16-s − 0.532·17-s − 1.05·18-s + 1.48·19-s + 1.78·21-s − 0.639·22-s + 0.528·23-s − 0.557·24-s + 0.769·27-s + 0.566·28-s − 1.75·29-s + 0.227·31-s − 0.176·32-s + 1.42·33-s + 0.376·34-s + 0.744·36-s + 1.84·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.684794270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.684794270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 - 2.53T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 - 6.19T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.80366694230644971267200281217, −7.48055651906683752018593183982, −6.80641808270507551526578347551, −5.77179584881438862264453518382, −4.88795858040285611245616278232, −4.00976060686471117412418566200, −3.37401466815977721440861306340, −2.48640336881630529349935287864, −1.77597565448545206966358290055, −1.05523977213570428593308749619,
1.05523977213570428593308749619, 1.77597565448545206966358290055, 2.48640336881630529349935287864, 3.37401466815977721440861306340, 4.00976060686471117412418566200, 4.88795858040285611245616278232, 5.77179584881438862264453518382, 6.80641808270507551526578347551, 7.48055651906683752018593183982, 7.80366694230644971267200281217
