| L(s) = 1 | − 3-s + 9-s − 11-s + 3·13-s + 2·17-s − 2·23-s − 27-s + 6·29-s − 11·31-s + 33-s − 4·37-s − 3·39-s + 2·41-s − 11·43-s + 6·47-s − 2·51-s + 4·53-s + 3·59-s − 8·61-s − 8·67-s + 2·69-s + 13·71-s − 3·73-s − 10·79-s + 81-s + 11·83-s − 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.485·17-s − 0.417·23-s − 0.192·27-s + 1.11·29-s − 1.97·31-s + 0.174·33-s − 0.657·37-s − 0.480·39-s + 0.312·41-s − 1.67·43-s + 0.875·47-s − 0.280·51-s + 0.549·53-s + 0.390·59-s − 1.02·61-s − 0.977·67-s + 0.240·69-s + 1.54·71-s − 0.351·73-s − 1.12·79-s + 1/9·81-s + 1.20·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227700274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227700274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.31911960813450, −12.71776442235002, −12.17808647254264, −12.02479152588878, −11.29451307538006, −10.81878807401711, −10.60470294269276, −9.938745285248452, −9.578566805091420, −8.840271408457736, −8.527408749069994, −7.917812766111895, −7.404100281211591, −6.886656202924525, −6.389813510662372, −5.835029575400827, −5.383270880747771, −4.969818716703187, −4.210652293749801, −3.745380956069263, −3.206588837780755, −2.487417553298260, −1.704980361558732, −1.233138049390022, −0.3491595584002073,
0.3491595584002073, 1.233138049390022, 1.704980361558732, 2.487417553298260, 3.206588837780755, 3.745380956069263, 4.210652293749801, 4.969818716703187, 5.383270880747771, 5.835029575400827, 6.389813510662372, 6.886656202924525, 7.404100281211591, 7.917812766111895, 8.527408749069994, 8.840271408457736, 9.578566805091420, 9.938745285248452, 10.60470294269276, 10.81878807401711, 11.29451307538006, 12.02479152588878, 12.17808647254264, 12.71776442235002, 13.31911960813450
