L(s) = 1 | + 3.25·3-s − 2.40·5-s + 4.10·7-s + 7.59·9-s − 5.28·11-s − 1.26·13-s − 7.83·15-s − 2.46·17-s + 4.09·19-s + 13.3·21-s + 0.252·23-s + 0.790·25-s + 14.9·27-s − 2.53·29-s − 0.649·31-s − 17.2·33-s − 9.88·35-s + 6.06·37-s − 4.12·39-s − 9.02·41-s + 8.38·43-s − 18.2·45-s + 10.6·47-s + 9.86·49-s − 8.03·51-s + 5.73·53-s + 12.7·55-s + ⋯ |
L(s) = 1 | + 1.87·3-s − 1.07·5-s + 1.55·7-s + 2.53·9-s − 1.59·11-s − 0.351·13-s − 2.02·15-s − 0.598·17-s + 0.940·19-s + 2.91·21-s + 0.0526·23-s + 0.158·25-s + 2.87·27-s − 0.471·29-s − 0.116·31-s − 2.99·33-s − 1.67·35-s + 0.997·37-s − 0.659·39-s − 1.41·41-s + 1.27·43-s − 2.72·45-s + 1.54·47-s + 1.40·49-s − 1.12·51-s + 0.787·53-s + 1.71·55-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
\(L(1)\) |
\(\approx\) |
\(4.037930575\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.037930575\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 + 5.28T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 - 0.252T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 0.649T + 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 - 8.38T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 2.38T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 8.81T + 97T^{2} \) |
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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
−7.75457641732350251308374392836, −7.61995534652868867922761368313, −7.00391764083270623120968460494, −5.40851662810727788074811964675, −4.85699406149773787878234919959, −4.08564467067917786221137513009, −3.56130057016169653074305859857, −2.44256387688218654203425990614, −2.24133549069415570771562174557, −0.917645150567266432434897641378,
0.917645150567266432434897641378, 2.24133549069415570771562174557, 2.44256387688218654203425990614, 3.56130057016169653074305859857, 4.08564467067917786221137513009, 4.85699406149773787878234919959, 5.40851662810727788074811964675, 7.00391764083270623120968460494, 7.61995534652868867922761368313, 7.75457641732350251308374392836