| L(s) = 1 | + 3.24·3-s + 0.470·5-s − 3.30·7-s + 7.55·9-s + 1.47·11-s − 2.83·13-s + 1.52·15-s + 6.71·17-s − 10.7·21-s + 0.470·23-s − 4.77·25-s + 14.8·27-s + 4.96·29-s + 6.74·31-s + 4.77·33-s − 1.55·35-s + 4.74·37-s − 9.21·39-s − 0.501·41-s + 3.89·43-s + 3.55·45-s + 11.9·47-s + 3.94·49-s + 21.8·51-s − 4.71·53-s + 0.692·55-s + 7.24·59-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 0.210·5-s − 1.25·7-s + 2.51·9-s + 0.443·11-s − 0.786·13-s + 0.394·15-s + 1.62·17-s − 2.34·21-s + 0.0981·23-s − 0.955·25-s + 2.84·27-s + 0.922·29-s + 1.21·31-s + 0.831·33-s − 0.263·35-s + 0.780·37-s − 1.47·39-s − 0.0783·41-s + 0.594·43-s + 0.530·45-s + 1.73·47-s + 0.563·49-s + 3.05·51-s − 0.648·53-s + 0.0933·55-s + 0.943·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.673907061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.673907061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 0.470T + 5T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 23 | \( 1 - 0.470T + 23T^{2} \) |
| 29 | \( 1 - 4.96T + 29T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + 0.501T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 - 7.24T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 0.633T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 3.94T + 73T^{2} \) |
| 79 | \( 1 + 8.83T + 79T^{2} \) |
| 83 | \( 1 - 9.14T + 83T^{2} \) |
| 89 | \( 1 - 2.95T + 89T^{2} \) |
| 97 | \( 1 - 0.0586T + 97T^{2} \) |
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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
−8.911699393471489820029160859027, −7.948167323758590584981162411391, −7.52889964029252082406694232326, −6.68189364739947068735242823568, −5.86072542263221445190859085501, −4.55047827049431742796931243021, −3.74198118677421519650237278523, −3.00279507739663639574548361722, −2.45766953413315737078433906698, −1.15897013414195697438185597817,
1.15897013414195697438185597817, 2.45766953413315737078433906698, 3.00279507739663639574548361722, 3.74198118677421519650237278523, 4.55047827049431742796931243021, 5.86072542263221445190859085501, 6.68189364739947068735242823568, 7.52889964029252082406694232326, 7.948167323758590584981162411391, 8.911699393471489820029160859027
