L(s) = 1 | + 3-s + 5-s + 5.12·7-s + 9-s − 5.12·11-s + 2·13-s + 15-s + 7.12·17-s − 4·19-s + 5.12·21-s − 23-s + 25-s + 27-s + 2·29-s − 5.12·33-s + 5.12·35-s − 7.12·37-s + 2·39-s + 2·41-s + 45-s + 8·47-s + 19.2·49-s + 7.12·51-s − 4.24·53-s − 5.12·55-s − 4·57-s + 14.2·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.93·7-s + 0.333·9-s − 1.54·11-s + 0.554·13-s + 0.258·15-s + 1.72·17-s − 0.917·19-s + 1.11·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s − 0.891·33-s + 0.865·35-s − 1.17·37-s + 0.320·39-s + 0.312·41-s + 0.149·45-s + 1.16·47-s + 2.74·49-s + 0.997·51-s − 0.583·53-s − 0.690·55-s − 0.529·57-s + 1.85·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.526786239\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.526786239\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 3.12T + 89T^{2} \) |
| 97 | \( 1 - 0.246T + 97T^{2} \) |
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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
−8.323520691567719437872140892020, −7.68276280539784889003063407873, −6.94596372305853404286142113924, −5.64927161199191280729974981851, −5.36658482453706172220422339344, −4.56711358443469277949494644551, −3.69057504907060679279500201741, −2.61370156504751435478077860978, −1.95503219694816842063462380477, −1.04911760106001196487966495165,
1.04911760106001196487966495165, 1.95503219694816842063462380477, 2.61370156504751435478077860978, 3.69057504907060679279500201741, 4.56711358443469277949494644551, 5.36658482453706172220422339344, 5.64927161199191280729974981851, 6.94596372305853404286142113924, 7.68276280539784889003063407873, 8.323520691567719437872140892020