| L(s) = 1 | − 0.979·2-s + 3-s − 1.04·4-s + 5-s − 0.979·6-s + 4.18·7-s + 2.97·8-s + 9-s − 0.979·10-s + 2.72·11-s − 1.04·12-s − 1.73·13-s − 4.09·14-s + 15-s − 0.833·16-s + 3.37·17-s − 0.979·18-s + 8.21·19-s − 1.04·20-s + 4.18·21-s − 2.66·22-s − 6.26·23-s + 2.97·24-s + 25-s + 1.70·26-s + 27-s − 4.35·28-s + ⋯ |
| L(s) = 1 | − 0.692·2-s + 0.577·3-s − 0.520·4-s + 0.447·5-s − 0.399·6-s + 1.58·7-s + 1.05·8-s + 0.333·9-s − 0.309·10-s + 0.821·11-s − 0.300·12-s − 0.482·13-s − 1.09·14-s + 0.258·15-s − 0.208·16-s + 0.818·17-s − 0.230·18-s + 1.88·19-s − 0.232·20-s + 0.913·21-s − 0.568·22-s − 1.30·23-s + 0.607·24-s + 0.200·25-s + 0.333·26-s + 0.192·27-s − 0.823·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.408824045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408824045\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.979T + 2T^{2} \) |
| 7 | \( 1 - 4.18T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 1.73T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 19 | \( 1 - 8.21T + 19T^{2} \) |
| 23 | \( 1 + 6.26T + 23T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 31 | \( 1 - 7.14T + 31T^{2} \) |
| 37 | \( 1 + 5.21T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 4.53T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 + 8.32T + 67T^{2} \) |
| 71 | \( 1 - 2.23T + 71T^{2} \) |
| 73 | \( 1 - 5.71T + 73T^{2} \) |
| 79 | \( 1 - 4.00T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 6.27T + 97T^{2} \) |
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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
−8.009807205286133570017134145982, −7.72468622590074558805317266118, −7.05224415354339017022614747013, −5.75388708358965513420665479124, −5.20830770705283322224152907223, −4.40028651941648651275689100513, −3.77116407485319654073985656398, −2.55246269764038632916713891783, −1.55026145483081030260525500651, −1.03161780042128466878301316212,
1.03161780042128466878301316212, 1.55026145483081030260525500651, 2.55246269764038632916713891783, 3.77116407485319654073985656398, 4.40028651941648651275689100513, 5.20830770705283322224152907223, 5.75388708358965513420665479124, 7.05224415354339017022614747013, 7.72468622590074558805317266118, 8.009807205286133570017134145982
