| L(s) = 1 | − 2.29·2-s + 3.40·3-s + 3.26·4-s + 3.88·5-s − 7.80·6-s + 2.49·7-s − 2.90·8-s + 8.57·9-s − 8.90·10-s − 11-s + 11.1·12-s − 3.83·13-s − 5.72·14-s + 13.2·15-s + 0.134·16-s − 0.384·17-s − 19.6·18-s − 3.93·19-s + 12.6·20-s + 8.48·21-s + 2.29·22-s − 1.53·23-s − 9.88·24-s + 10.0·25-s + 8.80·26-s + 18.9·27-s + 8.14·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.96·3-s + 1.63·4-s + 1.73·5-s − 3.18·6-s + 0.942·7-s − 1.02·8-s + 2.85·9-s − 2.81·10-s − 0.301·11-s + 3.20·12-s − 1.06·13-s − 1.52·14-s + 3.40·15-s + 0.0336·16-s − 0.0933·17-s − 4.63·18-s − 0.902·19-s + 2.83·20-s + 1.85·21-s + 0.489·22-s − 0.320·23-s − 2.01·24-s + 2.01·25-s + 1.72·26-s + 3.64·27-s + 1.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.142478143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.142478143\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 - 3.88T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 13 | \( 1 + 3.83T + 13T^{2} \) |
| 17 | \( 1 + 0.384T + 17T^{2} \) |
| 19 | \( 1 + 3.93T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 - 4.79T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 + 4.80T + 43T^{2} \) |
| 47 | \( 1 - 1.56T + 47T^{2} \) |
| 53 | \( 1 + 1.13T + 53T^{2} \) |
| 59 | \( 1 + 9.39T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 3.81T + 67T^{2} \) |
| 71 | \( 1 - 7.77T + 71T^{2} \) |
| 73 | \( 1 + 2.25T + 73T^{2} \) |
| 79 | \( 1 - 0.175T + 79T^{2} \) |
| 83 | \( 1 - 7.26T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 1.38T + 97T^{2} \) |
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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
−8.285204480034355553755768666471, −7.82796422285227622840717878901, −6.91728469423411900148947025538, −6.50897020539812285789393244933, −5.09546132568087115113702188428, −4.45954542148192344664074534752, −3.01873757474617883600651783248, −2.20302954941412483309089139313, −2.04644143380738079080567330222, −1.17701161606289660960294870952,
1.17701161606289660960294870952, 2.04644143380738079080567330222, 2.20302954941412483309089139313, 3.01873757474617883600651783248, 4.45954542148192344664074534752, 5.09546132568087115113702188428, 6.50897020539812285789393244933, 6.91728469423411900148947025538, 7.82796422285227622840717878901, 8.285204480034355553755768666471
