| L(s) = 1 | + 2-s − 3-s + 4-s − 3.26·5-s − 6-s + 0.499·7-s + 8-s + 9-s − 3.26·10-s + 3.54·11-s − 12-s − 3.54·13-s + 0.499·14-s + 3.26·15-s + 16-s − 4.88·17-s + 18-s − 19-s − 3.26·20-s − 0.499·21-s + 3.54·22-s + 9.10·23-s − 24-s + 5.67·25-s − 3.54·26-s − 27-s + 0.499·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.46·5-s − 0.408·6-s + 0.188·7-s + 0.353·8-s + 0.333·9-s − 1.03·10-s + 1.07·11-s − 0.288·12-s − 0.982·13-s + 0.133·14-s + 0.843·15-s + 0.250·16-s − 1.18·17-s + 0.235·18-s − 0.229·19-s − 0.730·20-s − 0.108·21-s + 0.756·22-s + 1.89·23-s − 0.204·24-s + 1.13·25-s − 0.694·26-s − 0.192·27-s + 0.0943·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 + 3.26T + 5T^{2} \) |
| 7 | \( 1 - 0.499T + 7T^{2} \) |
| 11 | \( 1 - 3.54T + 11T^{2} \) |
| 13 | \( 1 + 3.54T + 13T^{2} \) |
| 17 | \( 1 + 4.88T + 17T^{2} \) |
| 23 | \( 1 - 9.10T + 23T^{2} \) |
| 29 | \( 1 + 1.04T + 29T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 + 3.57T + 43T^{2} \) |
| 47 | \( 1 + 0.537T + 47T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 0.956T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.41061057669927687916466845695, −7.00021848532321308723071966396, −6.42945113154790614672365865956, −5.39500968657206444331796802341, −4.59757841571376829324165142621, −4.28663156787458449878313466949, −3.45153043860315239159494360650, −2.52277440492817133418734942649, −1.24195684447995353490767402546, 0,
1.24195684447995353490767402546, 2.52277440492817133418734942649, 3.45153043860315239159494360650, 4.28663156787458449878313466949, 4.59757841571376829324165142621, 5.39500968657206444331796802341, 6.42945113154790614672365865956, 7.00021848532321308723071966396, 7.41061057669927687916466845695
