L(s) = 1 | − 2.95·3-s + 0.296·5-s − 1.50·7-s + 5.72·9-s − 1.08·11-s − 3.03·13-s − 0.874·15-s − 1.39·17-s − 6.49·19-s + 4.43·21-s − 6.01·23-s − 4.91·25-s − 8.03·27-s + 2.16·29-s − 4.62·31-s + 3.19·33-s − 0.444·35-s + 3.14·37-s + 8.95·39-s − 3.26·41-s + 3.55·43-s + 1.69·45-s − 0.543·47-s − 4.74·49-s + 4.10·51-s − 9.68·53-s − 0.319·55-s + ⋯ |
L(s) = 1 | − 1.70·3-s + 0.132·5-s − 0.567·7-s + 1.90·9-s − 0.325·11-s − 0.841·13-s − 0.225·15-s − 0.337·17-s − 1.49·19-s + 0.966·21-s − 1.25·23-s − 0.982·25-s − 1.54·27-s + 0.401·29-s − 0.830·31-s + 0.555·33-s − 0.0750·35-s + 0.517·37-s + 1.43·39-s − 0.510·41-s + 0.542·43-s + 0.252·45-s − 0.0792·47-s − 0.678·49-s + 0.574·51-s − 1.33·53-s − 0.0431·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1006654030\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1006654030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 2.95T + 3T^{2} \) |
| 5 | \( 1 - 0.296T + 5T^{2} \) |
| 7 | \( 1 + 1.50T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 + 6.01T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 + 0.543T + 47T^{2} \) |
| 53 | \( 1 + 9.68T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 - 6.05T + 61T^{2} \) |
| 67 | \( 1 - 8.90T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 2.05T + 73T^{2} \) |
| 79 | \( 1 + 9.79T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 + 9.53T + 89T^{2} \) |
| 97 | \( 1 - 0.238T + 97T^{2} \) |
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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
−7.63591330110459322157191798603, −6.92880780847911236233220472075, −6.19535322926220828874578436028, −5.98702801607906444104732594064, −5.07973922143623741724674098577, −4.48881588171793215448381734762, −3.78495702066751164749183392002, −2.49557371897256557317259844476, −1.64728289228886557235302459312, −0.16640334111048308022118797027,
0.16640334111048308022118797027, 1.64728289228886557235302459312, 2.49557371897256557317259844476, 3.78495702066751164749183392002, 4.48881588171793215448381734762, 5.07973922143623741724674098577, 5.98702801607906444104732594064, 6.19535322926220828874578436028, 6.92880780847911236233220472075, 7.63591330110459322157191798603