L(s) = 1 | + 0.747·2-s − 2.67·3-s − 1.44·4-s − 5-s − 2.00·6-s − 3.83·7-s − 2.57·8-s + 4.16·9-s − 0.747·10-s − 2.47·11-s + 3.85·12-s − 5.59·13-s − 2.86·14-s + 2.67·15-s + 0.960·16-s − 5.49·17-s + 3.11·18-s + 2.62·19-s + 1.44·20-s + 10.2·21-s − 1.85·22-s − 7.47·23-s + 6.88·24-s + 25-s − 4.18·26-s − 3.12·27-s + 5.53·28-s + ⋯ |
L(s) = 1 | + 0.528·2-s − 1.54·3-s − 0.720·4-s − 0.447·5-s − 0.816·6-s − 1.45·7-s − 0.909·8-s + 1.38·9-s − 0.236·10-s − 0.746·11-s + 1.11·12-s − 1.55·13-s − 0.766·14-s + 0.691·15-s + 0.240·16-s − 1.33·17-s + 0.734·18-s + 0.601·19-s + 0.322·20-s + 2.24·21-s − 0.394·22-s − 1.55·23-s + 1.40·24-s + 0.200·25-s − 0.819·26-s − 0.601·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05531704981\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05531704981\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 0.747T + 2T^{2} \) |
| 3 | \( 1 + 2.67T + 3T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 + 0.734T + 29T^{2} \) |
| 31 | \( 1 - 3.92T + 31T^{2} \) |
| 37 | \( 1 + 4.79T + 37T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 - 2.88T + 43T^{2} \) |
| 47 | \( 1 - 6.57T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 2.58T + 71T^{2} \) |
| 73 | \( 1 - 7.39T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 9.72T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 8.19T + 97T^{2} \) |
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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
−9.934334525038049995718598627306, −9.198500122840218795249673325154, −7.958700184512556408450700415031, −6.89951103054841712381458839052, −6.28003587155177788927944066502, −5.39133204230915789419388735862, −4.77129562212859620317044334125, −3.89141752804640688272405535776, −2.69095571216717052793317167303, −0.16225406526361929127734499919,
0.16225406526361929127734499919, 2.69095571216717052793317167303, 3.89141752804640688272405535776, 4.77129562212859620317044334125, 5.39133204230915789419388735862, 6.28003587155177788927944066502, 6.89951103054841712381458839052, 7.958700184512556408450700415031, 9.198500122840218795249673325154, 9.934334525038049995718598627306