| L(s) = 1 | − 3-s − 0.561·5-s + 7-s + 9-s − 2.56·11-s + 13-s + 0.561·15-s + 5.68·17-s − 7.68·19-s − 21-s + 1.43·23-s − 4.68·25-s − 27-s − 5.68·29-s + 10.2·31-s + 2.56·33-s − 0.561·35-s − 3.43·37-s − 39-s + 7.12·41-s + 10.5·43-s − 0.561·45-s + 49-s − 5.68·51-s − 4.24·53-s + 1.43·55-s + 7.68·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.251·5-s + 0.377·7-s + 0.333·9-s − 0.772·11-s + 0.277·13-s + 0.144·15-s + 1.37·17-s − 1.76·19-s − 0.218·21-s + 0.299·23-s − 0.936·25-s − 0.192·27-s − 1.05·29-s + 1.84·31-s + 0.445·33-s − 0.0949·35-s − 0.565·37-s − 0.160·39-s + 1.11·41-s + 1.61·43-s − 0.0837·45-s + 0.142·49-s − 0.796·51-s − 0.583·53-s + 0.193·55-s + 1.01·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.302182589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302182589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 0.561T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.174758195477873674902437036817, −7.74261461489559210390978058711, −6.94121701775449860723157264232, −5.95372204930711142083955569183, −5.60285197586773223982111565410, −4.56682322034093923531413724618, −4.01190687968821877763013966021, −2.89050009681018490109509802532, −1.88056533104372271822735531205, −0.65933283836549223731925450048,
0.65933283836549223731925450048, 1.88056533104372271822735531205, 2.89050009681018490109509802532, 4.01190687968821877763013966021, 4.56682322034093923531413724618, 5.60285197586773223982111565410, 5.95372204930711142083955569183, 6.94121701775449860723157264232, 7.74261461489559210390978058711, 8.174758195477873674902437036817
