| L(s) = 1 | − 2-s + 3.05·3-s + 4-s − 3.05·6-s − 8-s + 6.31·9-s + 5.31·11-s + 3.05·12-s + 3.27·13-s + 16-s − 7.29·17-s − 6.31·18-s + 1.63·19-s − 5.31·22-s + 8.63·23-s − 3.05·24-s − 3.27·26-s + 10.1·27-s − 2.63·29-s − 5.65·31-s − 32-s + 16.2·33-s + 7.29·34-s + 6.31·36-s + 4.63·37-s − 1.63·38-s + 10·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.76·3-s + 0.5·4-s − 1.24·6-s − 0.353·8-s + 2.10·9-s + 1.60·11-s + 0.881·12-s + 0.908·13-s + 0.250·16-s − 1.76·17-s − 1.48·18-s + 0.375·19-s − 1.13·22-s + 1.80·23-s − 0.623·24-s − 0.642·26-s + 1.94·27-s − 0.488·29-s − 1.01·31-s − 0.176·32-s + 2.82·33-s + 1.25·34-s + 1.05·36-s + 0.761·37-s − 0.265·38-s + 1.60·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.895307418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.895307418\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.05T + 3T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 - 1.63T + 19T^{2} \) |
| 23 | \( 1 - 8.63T + 23T^{2} \) |
| 29 | \( 1 + 2.63T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 4.63T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + 6.10T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 1.86T + 59T^{2} \) |
| 61 | \( 1 - 6.10T + 61T^{2} \) |
| 67 | \( 1 + 1.63T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 7.29T + 73T^{2} \) |
| 79 | \( 1 - 1.36T + 79T^{2} \) |
| 83 | \( 1 + 8.26T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.955958042653761061319738874181, −8.541624647693002482722669577820, −7.53172753950055716002076567908, −6.90647096547688923688782395980, −6.26835048139795830053082871328, −4.68012738724509960551107730283, −3.77565506100005475186410293347, −3.13981258706124937170125490493, −2.04117898929986723485972763240, −1.25992640437677168354135482499,
1.25992640437677168354135482499, 2.04117898929986723485972763240, 3.13981258706124937170125490493, 3.77565506100005475186410293347, 4.68012738724509960551107730283, 6.26835048139795830053082871328, 6.90647096547688923688782395980, 7.53172753950055716002076567908, 8.541624647693002482722669577820, 8.955958042653761061319738874181
