| L(s) = 1 | − 1.17·3-s − 1.62·9-s + 6.62·11-s − 5.64·13-s − 7.99·17-s + 5.42·27-s − 0.623·29-s − 7.77·33-s + 6.62·39-s + 12.4·47-s + 9.37·51-s − 12·71-s + 13.4·73-s + 15.8·79-s − 1.49·81-s − 8.94·83-s + 0.731·87-s − 12.6·97-s − 10.7·99-s + 7.77·103-s − 9.87·109-s + 9.16·117-s + ⋯ |
| L(s) = 1 | − 0.677·3-s − 0.541·9-s + 1.99·11-s − 1.56·13-s − 1.93·17-s + 1.04·27-s − 0.115·29-s − 1.35·33-s + 1.06·39-s + 1.81·47-s + 1.31·51-s − 1.42·71-s + 1.57·73-s + 1.78·79-s − 0.165·81-s − 0.981·83-s + 0.0784·87-s − 1.28·97-s − 1.08·99-s + 0.765·103-s − 0.945·109-s + 0.847·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.073756859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073756859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.17T + 3T^{2} \) |
| 11 | \( 1 - 6.62T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 7.99T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 0.623T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.386180340092799930545843894106, −7.28390319063059513525944903807, −6.73491898727523643501237226985, −6.22179989661258799682394320593, −5.34648412163491855297776520707, −4.53490359577722223184028317113, −3.98856851767641320321434589048, −2.76352904329748997353639680725, −1.90613398862850428366013765502, −0.57441146660435581255081164576,
0.57441146660435581255081164576, 1.90613398862850428366013765502, 2.76352904329748997353639680725, 3.98856851767641320321434589048, 4.53490359577722223184028317113, 5.34648412163491855297776520707, 6.22179989661258799682394320593, 6.73491898727523643501237226985, 7.28390319063059513525944903807, 8.386180340092799930545843894106
