| L(s) = 1 | − 2.56·2-s − 0.903·3-s + 4.59·4-s + 2.32·6-s − 2.16·7-s − 6.65·8-s − 2.18·9-s − 11-s − 4.15·12-s + 4.47·13-s + 5.56·14-s + 7.90·16-s − 2.71·17-s + 5.60·18-s − 19-s + 1.96·21-s + 2.56·22-s + 2.85·23-s + 6.01·24-s − 11.4·26-s + 4.68·27-s − 9.95·28-s − 2.17·29-s − 6.53·31-s − 6.97·32-s + 0.903·33-s + 6.96·34-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.521·3-s + 2.29·4-s + 0.947·6-s − 0.819·7-s − 2.35·8-s − 0.727·9-s − 0.301·11-s − 1.19·12-s + 1.24·13-s + 1.48·14-s + 1.97·16-s − 0.657·17-s + 1.32·18-s − 0.229·19-s + 0.427·21-s + 0.547·22-s + 0.595·23-s + 1.22·24-s − 2.25·26-s + 0.901·27-s − 1.88·28-s − 0.404·29-s − 1.17·31-s − 1.23·32-s + 0.157·33-s + 1.19·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 + 0.903T + 3T^{2} \) |
| 7 | \( 1 + 2.16T + 7T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2.71T + 17T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 + 6.53T + 31T^{2} \) |
| 37 | \( 1 + 0.861T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 8.75T + 43T^{2} \) |
| 47 | \( 1 - 0.665T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 - 5.59T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 5.99T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 5.74T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 4.83T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.047671005344323087393034518056, −7.19313860426900126595300317752, −6.57730387978970198239422206747, −6.03474741831213720905638160328, −5.30087256261730099299649753396, −3.85884512913562412324710752762, −2.95458918729745372071998955628, −2.09331058939856502589404223913, −0.916759691176279815755377172468, 0,
0.916759691176279815755377172468, 2.09331058939856502589404223913, 2.95458918729745372071998955628, 3.85884512913562412324710752762, 5.30087256261730099299649753396, 6.03474741831213720905638160328, 6.57730387978970198239422206747, 7.19313860426900126595300317752, 8.047671005344323087393034518056
