| L(s) = 1 | + 1.61·3-s + 0.618·5-s + 7-s − 0.381·9-s + 0.763·11-s + 1.23·13-s + 1.00·15-s + 17-s + 8.47·19-s + 1.61·21-s + 7.70·23-s − 4.61·25-s − 5.47·27-s − 5.70·29-s − 6.32·31-s + 1.23·33-s + 0.618·35-s + 0.472·37-s + 2.00·39-s − 0.0901·41-s + 12.0·43-s − 0.236·45-s + 8.47·47-s + 49-s + 1.61·51-s + 10.7·53-s + 0.472·55-s + ⋯ |
| L(s) = 1 | + 0.934·3-s + 0.276·5-s + 0.377·7-s − 0.127·9-s + 0.230·11-s + 0.342·13-s + 0.258·15-s + 0.242·17-s + 1.94·19-s + 0.353·21-s + 1.60·23-s − 0.923·25-s − 1.05·27-s − 1.05·29-s − 1.13·31-s + 0.215·33-s + 0.104·35-s + 0.0776·37-s + 0.320·39-s − 0.0140·41-s + 1.84·43-s − 0.0351·45-s + 1.23·47-s + 0.142·49-s + 0.226·51-s + 1.48·53-s + 0.0636·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.780679386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.780679386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 19 | \( 1 - 8.47T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 0.0901T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 8.47T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.32T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.251494114526931680146744358608, −8.557481013191392824325509638094, −7.52270453110405666428798929251, −7.27223624438057612688824540281, −5.76677596764495562855844723021, −5.37840242461091345403321708712, −4.02144685807586174488539976853, −3.26944270530189560424564744998, −2.35049339893568582670634373650, −1.17578664884106330615335621997,
1.17578664884106330615335621997, 2.35049339893568582670634373650, 3.26944270530189560424564744998, 4.02144685807586174488539976853, 5.37840242461091345403321708712, 5.76677596764495562855844723021, 7.27223624438057612688824540281, 7.52270453110405666428798929251, 8.557481013191392824325509638094, 9.251494114526931680146744358608
