L(s) = 1 | + 0.933·2-s − 3-s − 1.12·4-s − 5-s − 0.933·6-s + 2.04·7-s − 2.92·8-s + 9-s − 0.933·10-s + 1.12·12-s + 1.44·13-s + 1.90·14-s + 15-s − 0.469·16-s − 0.867·17-s + 0.933·18-s − 3.12·19-s + 1.12·20-s − 2.04·21-s − 4.70·23-s + 2.92·24-s + 25-s + 1.35·26-s − 27-s − 2.30·28-s − 2.03·29-s + 0.933·30-s + ⋯ |
L(s) = 1 | + 0.660·2-s − 0.577·3-s − 0.564·4-s − 0.447·5-s − 0.381·6-s + 0.771·7-s − 1.03·8-s + 0.333·9-s − 0.295·10-s + 0.325·12-s + 0.401·13-s + 0.509·14-s + 0.258·15-s − 0.117·16-s − 0.210·17-s + 0.220·18-s − 0.717·19-s + 0.252·20-s − 0.445·21-s − 0.981·23-s + 0.596·24-s + 0.200·25-s + 0.265·26-s − 0.192·27-s − 0.435·28-s − 0.378·29-s + 0.170·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.445939289\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.445939289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.933T + 2T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 + 0.867T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 - 0.805T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 7.21T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 8.76T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 6.14T + 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 + 1.93T + 97T^{2} \) |
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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
−9.224994025827226373530013479040, −8.346771050865323269795751276658, −7.84605478710161565382743138994, −6.60133654974004880197596118106, −5.94383431608588669416431329664, −5.05449336978319709353830316699, −4.35763858972027751852824011858, −3.76332371883082025486930046560, −2.37145179638812239777919654933, −0.77167416610718969447678728636,
0.77167416610718969447678728636, 2.37145179638812239777919654933, 3.76332371883082025486930046560, 4.35763858972027751852824011858, 5.05449336978319709353830316699, 5.94383431608588669416431329664, 6.60133654974004880197596118106, 7.84605478710161565382743138994, 8.346771050865323269795751276658, 9.224994025827226373530013479040