| L(s) = 1 | + 5.42·2-s + 21.4·4-s + 7.70·5-s + 15.0·7-s + 73.1·8-s + 41.8·10-s − 2.51·11-s + 81.5·14-s + 225.·16-s + 2.06·17-s + 94.4·19-s + 165.·20-s − 13.6·22-s − 35.8·23-s − 65.5·25-s + 322.·28-s + 140.·29-s − 264.·31-s + 637.·32-s + 11.2·34-s + 115.·35-s − 256.·37-s + 512.·38-s + 563.·40-s + 394.·41-s + 256.·43-s − 53.8·44-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.68·4-s + 0.689·5-s + 0.811·7-s + 3.23·8-s + 1.32·10-s − 0.0688·11-s + 1.55·14-s + 3.51·16-s + 0.0294·17-s + 1.14·19-s + 1.85·20-s − 0.132·22-s − 0.325·23-s − 0.524·25-s + 2.17·28-s + 0.897·29-s − 1.52·31-s + 3.51·32-s + 0.0565·34-s + 0.559·35-s − 1.14·37-s + 2.18·38-s + 2.22·40-s + 1.50·41-s + 0.910·43-s − 0.184·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.74502445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.74502445\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5.42T + 8T^{2} \) |
| 5 | \( 1 - 7.70T + 125T^{2} \) |
| 7 | \( 1 - 15.0T + 343T^{2} \) |
| 11 | \( 1 + 2.51T + 1.33e3T^{2} \) |
| 17 | \( 1 - 2.06T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 35.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 256.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 394.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 415.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 504.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 120.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 410.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 17.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 174.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 963.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 477.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−9.174605514785093062593583265324, −7.86163040855805096274594834515, −7.28547233245804920444774200636, −6.28591462845819080259477569057, −5.58192925770276342964094674163, −5.03060836661671001846314461494, −4.14276291233204906113291370499, −3.20990719134367834714974928326, −2.21513335482104857415666995430, −1.40809029777713015201956204620,
1.40809029777713015201956204620, 2.21513335482104857415666995430, 3.20990719134367834714974928326, 4.14276291233204906113291370499, 5.03060836661671001846314461494, 5.58192925770276342964094674163, 6.28591462845819080259477569057, 7.28547233245804920444774200636, 7.86163040855805096274594834515, 9.174605514785093062593583265324
