| L(s) = 1 | − 2.64·2-s − 1.66·3-s + 4.97·4-s + 4.41·6-s + 2.26·7-s − 7.86·8-s − 0.211·9-s − 4.12·11-s − 8.30·12-s − 5.91·13-s − 5.97·14-s + 10.8·16-s + 1.43·17-s + 0.558·18-s + 0.204·19-s − 3.77·21-s + 10.8·22-s − 6.71·23-s + 13.1·24-s + 15.6·26-s + 5.36·27-s + 11.2·28-s − 1.38·29-s − 3.06·31-s − 12.8·32-s + 6.88·33-s − 3.78·34-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.964·3-s + 2.48·4-s + 1.80·6-s + 0.855·7-s − 2.77·8-s − 0.0705·9-s − 1.24·11-s − 2.39·12-s − 1.63·13-s − 1.59·14-s + 2.70·16-s + 0.347·17-s + 0.131·18-s + 0.0469·19-s − 0.824·21-s + 2.32·22-s − 1.40·23-s + 2.67·24-s + 3.06·26-s + 1.03·27-s + 2.12·28-s − 0.257·29-s − 0.550·31-s − 2.26·32-s + 1.19·33-s − 0.649·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05121802575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05121802575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 + 1.66T + 3T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 0.204T + 19T^{2} \) |
| 23 | \( 1 + 6.71T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 + 9.92T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 + 2.85T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 4.56T + 67T^{2} \) |
| 71 | \( 1 - 0.472T + 71T^{2} \) |
| 73 | \( 1 - 3.38T + 73T^{2} \) |
| 79 | \( 1 + 4.84T + 79T^{2} \) |
| 83 | \( 1 - 4.30T + 83T^{2} \) |
| 89 | \( 1 - 2.35T + 89T^{2} \) |
| 97 | \( 1 - 6.51T + 97T^{2} \) |
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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
−8.049078217320595978134027090900, −7.59228122870096597629766218731, −6.89099679186250790373473578490, −6.12783569490865563624141759771, −5.25012316057132433004386833190, −4.88345477576878011717062492003, −3.21275487367425106446262246947, −2.27031083080948222196582102388, −1.61015601393560846889839201669, −0.15907883203981225671269895138,
0.15907883203981225671269895138, 1.61015601393560846889839201669, 2.27031083080948222196582102388, 3.21275487367425106446262246947, 4.88345477576878011717062492003, 5.25012316057132433004386833190, 6.12783569490865563624141759771, 6.89099679186250790373473578490, 7.59228122870096597629766218731, 8.049078217320595978134027090900
