L(s) = 1 | − 0.409·2-s − 1.91·3-s − 1.83·4-s + 0.785·6-s − 3.66·7-s + 1.56·8-s + 0.682·9-s + 5.60·11-s + 3.51·12-s + 4.56·13-s + 1.49·14-s + 3.02·16-s + 6.06·17-s − 0.279·18-s − 3.67·19-s + 7.02·21-s − 2.29·22-s + 7.36·23-s − 3.01·24-s − 1.86·26-s + 4.44·27-s + 6.70·28-s − 1.37·29-s + 2.62·31-s − 4.37·32-s − 10.7·33-s − 2.48·34-s + ⋯ |
L(s) = 1 | − 0.289·2-s − 1.10·3-s − 0.916·4-s + 0.320·6-s − 1.38·7-s + 0.554·8-s + 0.227·9-s + 1.68·11-s + 1.01·12-s + 1.26·13-s + 0.400·14-s + 0.755·16-s + 1.47·17-s − 0.0658·18-s − 0.843·19-s + 1.53·21-s − 0.489·22-s + 1.53·23-s − 0.614·24-s − 0.366·26-s + 0.855·27-s + 1.26·28-s − 0.255·29-s + 0.471·31-s − 0.773·32-s − 1.87·33-s − 0.425·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9358942157\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9358942157\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 0.409T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 7 | \( 1 + 3.66T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 6.06T + 17T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 - 8.36T + 43T^{2} \) |
| 47 | \( 1 - 8.91T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.04T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 4.63T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 0.0114T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.295483515141071351151781236980, −7.07946330191002580724190371975, −6.62729040649800079556546825594, −5.83892660322075353607732235786, −5.52353537340499901978460362840, −4.33051948001064749362387464871, −3.79056689654946843695218767569, −3.05282220071781270040170741404, −1.20425653825144113952383106775, −0.70135469423396928413827311001,
0.70135469423396928413827311001, 1.20425653825144113952383106775, 3.05282220071781270040170741404, 3.79056689654946843695218767569, 4.33051948001064749362387464871, 5.52353537340499901978460362840, 5.83892660322075353607732235786, 6.62729040649800079556546825594, 7.07946330191002580724190371975, 8.295483515141071351151781236980