| L(s) = 1 | − 0.363·3-s − 7-s − 2.86·9-s + 5.14·11-s + 4.64·13-s − 3.86·17-s − 0.778·19-s + 0.363·21-s − 5.00·23-s + 2.13·27-s + 9.42·29-s + 4.72·31-s − 1.86·33-s − 6·37-s − 1.68·39-s − 1.00·41-s + 7.00·43-s + 11.4·47-s + 49-s + 1.40·51-s − 7.55·53-s + 0.282·57-s + 12.5·59-s + 11.5·61-s + 2.86·63-s + 11.7·67-s + 1.82·69-s + ⋯ |
| L(s) = 1 | − 0.209·3-s − 0.377·7-s − 0.955·9-s + 1.55·11-s + 1.28·13-s − 0.938·17-s − 0.178·19-s + 0.0792·21-s − 1.04·23-s + 0.410·27-s + 1.74·29-s + 0.848·31-s − 0.325·33-s − 0.986·37-s − 0.270·39-s − 0.157·41-s + 1.06·43-s + 1.66·47-s + 0.142·49-s + 0.196·51-s − 1.03·53-s + 0.0374·57-s + 1.62·59-s + 1.47·61-s + 0.361·63-s + 1.43·67-s + 0.219·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.513456989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513456989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 3 | \( 1 + 0.363T + 3T^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 19 | \( 1 + 0.778T + 19T^{2} \) |
| 23 | \( 1 + 5.00T + 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 - 4.72T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 1.00T + 41T^{2} \) |
| 43 | \( 1 - 7.00T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
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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
−9.444875670669507094470764719879, −8.669383570223460816829694740894, −8.282315010477802170753222703561, −6.72984832126464717708769967367, −6.42686629260159248091551996064, −5.59342515765788872900937549542, −4.30087607381132231659988055947, −3.61680947352390500537034183457, −2.38814431097515511130754157654, −0.921380942318043290863832812987,
0.921380942318043290863832812987, 2.38814431097515511130754157654, 3.61680947352390500537034183457, 4.30087607381132231659988055947, 5.59342515765788872900937549542, 6.42686629260159248091551996064, 6.72984832126464717708769967367, 8.282315010477802170753222703561, 8.669383570223460816829694740894, 9.444875670669507094470764719879
