| L(s) = 1 | − 2·2-s + 4·4-s − 18·5-s + 7·7-s − 8·8-s + 36·10-s + 72·11-s − 34·13-s − 14·14-s + 16·16-s − 6·17-s + 92·19-s − 72·20-s − 144·22-s + 180·23-s + 199·25-s + 68·26-s + 28·28-s + 114·29-s + 56·31-s − 32·32-s + 12·34-s − 126·35-s − 34·37-s − 184·38-s + 144·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.60·5-s + 0.377·7-s − 0.353·8-s + 1.13·10-s + 1.97·11-s − 0.725·13-s − 0.267·14-s + 1/4·16-s − 0.0856·17-s + 1.11·19-s − 0.804·20-s − 1.39·22-s + 1.63·23-s + 1.59·25-s + 0.512·26-s + 0.188·28-s + 0.729·29-s + 0.324·31-s − 0.176·32-s + 0.0605·34-s − 0.608·35-s − 0.151·37-s − 0.785·38-s + 0.569·40-s − 0.0228·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9753957869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9753957869\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 72 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 - 114 T + p^{3} T^{2} \) |
| 31 | \( 1 - 56 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 654 T + p^{3} T^{2} \) |
| 59 | \( 1 - 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 250 T + p^{3} T^{2} \) |
| 67 | \( 1 + 124 T + p^{3} T^{2} \) |
| 71 | \( 1 + 36 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1010 T + p^{3} T^{2} \) |
| 79 | \( 1 - 56 T + p^{3} T^{2} \) |
| 83 | \( 1 + 228 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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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
−12.40058010281919343398812057366, −11.69565419688594271897873652891, −11.10304505863400021591546278710, −9.544710333616174132186572453208, −8.628620988453936622031152570414, −7.54131034300571593229364097681, −6.73529943685552547976746836199, −4.67529839296481476756708143517, −3.35728318624127117214776553458, −0.982037875331416021358349447959,
0.982037875331416021358349447959, 3.35728318624127117214776553458, 4.67529839296481476756708143517, 6.73529943685552547976746836199, 7.54131034300571593229364097681, 8.628620988453936622031152570414, 9.544710333616174132186572453208, 11.10304505863400021591546278710, 11.69565419688594271897873652891, 12.40058010281919343398812057366
