L(s) = 1 | + 2.27·2-s + 3-s + 3.16·4-s − 5-s + 2.27·6-s − 0.748·7-s + 2.65·8-s + 9-s − 2.27·10-s − 1.97·11-s + 3.16·12-s + 1.77·13-s − 1.70·14-s − 15-s − 0.293·16-s − 3.85·17-s + 2.27·18-s − 7.07·19-s − 3.16·20-s − 0.748·21-s − 4.48·22-s − 5.50·23-s + 2.65·24-s + 25-s + 4.04·26-s + 27-s − 2.37·28-s + ⋯ |
L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.58·4-s − 0.447·5-s + 0.928·6-s − 0.282·7-s + 0.939·8-s + 0.333·9-s − 0.718·10-s − 0.595·11-s + 0.914·12-s + 0.493·13-s − 0.454·14-s − 0.258·15-s − 0.0734·16-s − 0.935·17-s + 0.535·18-s − 1.62·19-s − 0.708·20-s − 0.163·21-s − 0.957·22-s − 1.14·23-s + 0.542·24-s + 0.200·25-s + 0.792·26-s + 0.192·27-s − 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 7 | \( 1 + 0.748T + 7T^{2} \) |
| 11 | \( 1 + 1.97T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 + 5.50T + 23T^{2} \) |
| 29 | \( 1 + 6.46T + 29T^{2} \) |
| 31 | \( 1 + 4.25T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 - 4.12T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 - 1.96T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 - 0.603T + 97T^{2} \) |
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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
−7.50421558822187533904783065569, −6.84358672507980378529906567266, −6.18160752883844373486181227908, −5.49609078546902472591133434060, −4.63603391089672408580054045013, −3.88867652159527187644009361444, −3.63629721205649185796603735828, −2.45826033611238148302092518501, −2.01953032205407223871365972351, 0,
2.01953032205407223871365972351, 2.45826033611238148302092518501, 3.63629721205649185796603735828, 3.88867652159527187644009361444, 4.63603391089672408580054045013, 5.49609078546902472591133434060, 6.18160752883844373486181227908, 6.84358672507980378529906567266, 7.50421558822187533904783065569