L(s) = 1 | + 2.56·2-s + 4.56·4-s + 7-s + 6.56·8-s − 11-s − 0.438·13-s + 2.56·14-s + 7.68·16-s + 1.43·17-s + 3·19-s − 2.56·22-s + 6.56·23-s − 1.12·26-s + 4.56·28-s + 5.12·29-s + 4.68·31-s + 6.56·32-s + 3.68·34-s − 10.8·37-s + 7.68·38-s − 7.68·41-s + 4.68·43-s − 4.56·44-s + 16.8·46-s + 5.43·47-s − 6·49-s − 2·52-s + ⋯ |
L(s) = 1 | + 1.81·2-s + 2.28·4-s + 0.377·7-s + 2.31·8-s − 0.301·11-s − 0.121·13-s + 0.684·14-s + 1.92·16-s + 0.348·17-s + 0.688·19-s − 0.546·22-s + 1.36·23-s − 0.220·26-s + 0.862·28-s + 0.951·29-s + 0.841·31-s + 1.15·32-s + 0.631·34-s − 1.77·37-s + 1.24·38-s − 1.20·41-s + 0.714·43-s − 0.687·44-s + 2.47·46-s + 0.793·47-s − 0.857·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.070865211\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.070865211\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 + 0.438T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 2.12T + 97T^{2} \) |
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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
−8.791921948685192521070249813559, −7.896104149932438201226223139528, −7.03526116404815752596373588659, −6.53010569592905784051965803540, −5.37024186115513835272819271057, −5.13059508483270540789814980649, −4.22041469467659197613539409522, −3.27024468103184251892209335808, −2.62542065015625907329286861147, −1.38384986319681027676454808451,
1.38384986319681027676454808451, 2.62542065015625907329286861147, 3.27024468103184251892209335808, 4.22041469467659197613539409522, 5.13059508483270540789814980649, 5.37024186115513835272819271057, 6.53010569592905784051965803540, 7.03526116404815752596373588659, 7.896104149932438201226223139528, 8.791921948685192521070249813559