| L(s) = 1 | + 2-s + 2.78·3-s + 4-s − 2.54·5-s + 2.78·6-s + 7-s + 8-s + 4.75·9-s − 2.54·10-s − 3.33·11-s + 2.78·12-s + 14-s − 7.09·15-s + 16-s + 7.09·17-s + 4.75·18-s + 4.54·19-s − 2.54·20-s + 2.78·21-s − 3.33·22-s + 1.75·23-s + 2.78·24-s + 1.48·25-s + 4.90·27-s + 28-s + 7.57·29-s − 7.09·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.60·3-s + 0.5·4-s − 1.13·5-s + 1.13·6-s + 0.377·7-s + 0.353·8-s + 1.58·9-s − 0.804·10-s − 1.00·11-s + 0.804·12-s + 0.267·14-s − 1.83·15-s + 0.250·16-s + 1.71·17-s + 1.12·18-s + 1.04·19-s − 0.569·20-s + 0.607·21-s − 0.710·22-s + 0.366·23-s + 0.568·24-s + 0.296·25-s + 0.943·27-s + 0.188·28-s + 1.40·29-s − 1.29·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.443517710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.443517710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.78T + 3T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 - 3.33T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 + 0.785T + 61T^{2} \) |
| 67 | \( 1 - 5.75T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 9.33T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 7.75T + 97T^{2} \) |
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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
−8.604790132461501077063834889722, −8.091991888911470007528292489514, −7.62813201947342147133290941826, −7.03652147234687469958797172452, −5.61755442302797857081414674339, −4.78712864584460083805666763700, −3.93278108095486278834891160442, −3.14209466665128511441449739710, −2.71198815822727519510267754414, −1.25742315353533444839105475764,
1.25742315353533444839105475764, 2.71198815822727519510267754414, 3.14209466665128511441449739710, 3.93278108095486278834891160442, 4.78712864584460083805666763700, 5.61755442302797857081414674339, 7.03652147234687469958797172452, 7.62813201947342147133290941826, 8.091991888911470007528292489514, 8.604790132461501077063834889722
