L(s) = 1 | − 2.61·2-s − 2.61·3-s + 4.85·4-s − 5-s + 6.85·6-s + 0.236·7-s − 7.47·8-s + 3.85·9-s + 2.61·10-s − 4.61·11-s − 12.7·12-s − 5·13-s − 0.618·14-s + 2.61·15-s + 9.85·16-s − 6·17-s − 10.0·18-s − 4.85·20-s − 0.618·21-s + 12.0·22-s − 1.61·23-s + 19.5·24-s + 25-s + 13.0·26-s − 2.23·27-s + 1.14·28-s + 1.85·29-s + ⋯ |
L(s) = 1 | − 1.85·2-s − 1.51·3-s + 2.42·4-s − 0.447·5-s + 2.79·6-s + 0.0892·7-s − 2.64·8-s + 1.28·9-s + 0.827·10-s − 1.39·11-s − 3.66·12-s − 1.38·13-s − 0.165·14-s + 0.675·15-s + 2.46·16-s − 1.45·17-s − 2.37·18-s − 1.08·20-s − 0.134·21-s + 2.57·22-s − 0.337·23-s + 3.99·24-s + 0.200·25-s + 2.56·26-s − 0.430·27-s + 0.216·28-s + 0.344·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 23 | \( 1 + 1.61T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 - 1.85T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 2.85T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 + 0.0901T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−8.370958914666709128461865326637, −7.81080219051163735509560428592, −6.94453987324061569269805639883, −6.51909181562675383555232123008, −5.35145293167089716622401256017, −4.67976170456253772045163608928, −2.83467863339657275170986967195, −1.76903145686835238146365551917, 0, 0,
1.76903145686835238146365551917, 2.83467863339657275170986967195, 4.67976170456253772045163608928, 5.35145293167089716622401256017, 6.51909181562675383555232123008, 6.94453987324061569269805639883, 7.81080219051163735509560428592, 8.370958914666709128461865326637