L(s) = 1 | + 2.85·3-s + 0.827·5-s + 5.16·9-s + 5.71·11-s − 0.0340·13-s + 2.36·15-s − 4.23·17-s + 19-s + 4.89·23-s − 4.31·25-s + 6.17·27-s + 9.29·29-s − 2.53·31-s + 16.3·33-s + 4.58·37-s − 0.0972·39-s − 2·41-s + 9.61·43-s + 4.27·45-s − 11.3·47-s − 12.0·51-s + 3.43·53-s + 4.73·55-s + 2.85·57-s − 7.35·59-s + 2.49·61-s − 0.0281·65-s + ⋯ |
L(s) = 1 | + 1.64·3-s + 0.370·5-s + 1.72·9-s + 1.72·11-s − 0.00943·13-s + 0.610·15-s − 1.02·17-s + 0.229·19-s + 1.02·23-s − 0.862·25-s + 1.18·27-s + 1.72·29-s − 0.455·31-s + 2.84·33-s + 0.753·37-s − 0.0155·39-s − 0.312·41-s + 1.46·43-s + 0.636·45-s − 1.65·47-s − 1.69·51-s + 0.471·53-s + 0.638·55-s + 0.378·57-s − 0.957·59-s + 0.319·61-s − 0.00349·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.923994750\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.923994750\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.85T + 3T^{2} \) |
| 5 | \( 1 - 0.827T + 5T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 + 0.0340T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 - 4.58T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 9.61T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 3.43T + 53T^{2} \) |
| 59 | \( 1 + 7.35T + 59T^{2} \) |
| 61 | \( 1 - 2.49T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 - 0.751T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−8.008110816422012422727281036918, −7.26193332135365972439442855003, −6.64077368345259466565547078637, −6.03577564921452833172576841461, −4.75559332373637974660815777526, −4.18321580807067416901304850295, −3.45106202508690602210847041815, −2.72069388028561751606906667875, −1.92100403643489553231850075471, −1.12376992463193954143160693599,
1.12376992463193954143160693599, 1.92100403643489553231850075471, 2.72069388028561751606906667875, 3.45106202508690602210847041815, 4.18321580807067416901304850295, 4.75559332373637974660815777526, 6.03577564921452833172576841461, 6.64077368345259466565547078637, 7.26193332135365972439442855003, 8.008110816422012422727281036918