| L(s) = 1 | − 3·3-s − 5·5-s + 32·7-s + 9·9-s − 64·11-s + 6·13-s + 15·15-s + 38·17-s + 116·19-s − 96·21-s − 120·23-s + 25·25-s − 27·27-s + 122·29-s + 164·31-s + 192·33-s − 160·35-s − 146·37-s − 18·39-s − 238·41-s + 148·43-s − 45·45-s − 184·47-s + 681·49-s − 114·51-s − 470·53-s + 320·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.72·7-s + 1/3·9-s − 1.75·11-s + 0.128·13-s + 0.258·15-s + 0.542·17-s + 1.40·19-s − 0.997·21-s − 1.08·23-s + 1/5·25-s − 0.192·27-s + 0.781·29-s + 0.950·31-s + 1.01·33-s − 0.772·35-s − 0.648·37-s − 0.0739·39-s − 0.906·41-s + 0.524·43-s − 0.149·45-s − 0.571·47-s + 1.98·49-s − 0.313·51-s − 1.21·53-s + 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.688842026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688842026\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 122 T + p^{3} T^{2} \) |
| 31 | \( 1 - 164 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 238 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 + 184 T + p^{3} T^{2} \) |
| 53 | \( 1 + 470 T + p^{3} T^{2} \) |
| 59 | \( 1 - 216 T + p^{3} T^{2} \) |
| 61 | \( 1 + 806 T + p^{3} T^{2} \) |
| 67 | \( 1 - 732 T + p^{3} T^{2} \) |
| 71 | \( 1 - 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 638 T + p^{3} T^{2} \) |
| 79 | \( 1 - 596 T + p^{3} T^{2} \) |
| 83 | \( 1 - 884 T + p^{3} T^{2} \) |
| 89 | \( 1 - 930 T + p^{3} T^{2} \) |
| 97 | \( 1 - 322 T + p^{3} T^{2} \) |
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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
−9.938460894132747897857396785492, −8.496526431683263931960041255657, −7.86563623140392392564374489302, −7.44788265904948076192256868074, −6.01825301939854062741964293775, −5.04201457577229651875157116141, −4.75224787739857953859482080710, −3.28116982077714151943725928818, −1.95337281523667189826740675843, −0.73278881697470875416392467579,
0.73278881697470875416392467579, 1.95337281523667189826740675843, 3.28116982077714151943725928818, 4.75224787739857953859482080710, 5.04201457577229651875157116141, 6.01825301939854062741964293775, 7.44788265904948076192256868074, 7.86563623140392392564374489302, 8.496526431683263931960041255657, 9.938460894132747897857396785492
