| L(s) = 1 | + 0.593·3-s − 2.48·5-s − 3.58·7-s − 2.64·9-s + 0.309·11-s − 1.08·13-s − 1.47·15-s + 5.71·17-s + 1.82·19-s − 2.12·21-s + 1.16·25-s − 3.35·27-s − 6.00·29-s − 6.94·31-s + 0.183·33-s + 8.89·35-s − 9.89·37-s − 0.640·39-s − 6.59·41-s + 2.87·43-s + 6.57·45-s − 12.3·47-s + 5.84·49-s + 3.39·51-s + 8.28·53-s − 0.768·55-s + 1.08·57-s + ⋯ |
| L(s) = 1 | + 0.342·3-s − 1.11·5-s − 1.35·7-s − 0.882·9-s + 0.0933·11-s − 0.299·13-s − 0.380·15-s + 1.38·17-s + 0.418·19-s − 0.463·21-s + 0.232·25-s − 0.644·27-s − 1.11·29-s − 1.24·31-s + 0.0319·33-s + 1.50·35-s − 1.62·37-s − 0.102·39-s − 1.03·41-s + 0.438·43-s + 0.979·45-s − 1.80·47-s + 0.834·49-s + 0.475·51-s + 1.13·53-s − 0.103·55-s + 0.143·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5027904604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5027904604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.593T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 - 0.309T + 11T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 29 | \( 1 + 6.00T + 29T^{2} \) |
| 31 | \( 1 + 6.94T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 8.28T + 53T^{2} \) |
| 59 | \( 1 + 3.48T + 59T^{2} \) |
| 61 | \( 1 + 8.06T + 61T^{2} \) |
| 67 | \( 1 + 1.48T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 3.51T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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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
−7.71621881875641250937765293870, −7.26865075600966305390215631990, −6.49927511039037347448876036921, −5.65610173196431248882480363333, −5.14247959040935598100134177840, −3.88784170997975498009268927699, −3.39374439257315847392765351714, −3.06977813807475154309638983796, −1.80828526531255206640708514493, −0.32431183451493135195633374654,
0.32431183451493135195633374654, 1.80828526531255206640708514493, 3.06977813807475154309638983796, 3.39374439257315847392765351714, 3.88784170997975498009268927699, 5.14247959040935598100134177840, 5.65610173196431248882480363333, 6.49927511039037347448876036921, 7.26865075600966305390215631990, 7.71621881875641250937765293870
