| L(s) = 1 | + 1.71·3-s + 0.0212·5-s + 3.28·7-s − 0.0600·9-s + 11-s − 4.54·13-s + 0.0363·15-s + 1.30·17-s − 8.21·19-s + 5.63·21-s − 0.264·23-s − 4.99·25-s − 5.24·27-s + 3.21·29-s − 7.34·31-s + 1.71·33-s + 0.0697·35-s − 3.03·37-s − 7.78·39-s − 8.41·41-s − 2.49·43-s − 0.00127·45-s − 9.67·47-s + 3.80·49-s + 2.23·51-s + 0.773·53-s + 0.0212·55-s + ⋯ |
| L(s) = 1 | + 0.989·3-s + 0.00949·5-s + 1.24·7-s − 0.0200·9-s + 0.301·11-s − 1.25·13-s + 0.00939·15-s + 0.316·17-s − 1.88·19-s + 1.23·21-s − 0.0551·23-s − 0.999·25-s − 1.00·27-s + 0.597·29-s − 1.31·31-s + 0.298·33-s + 0.0117·35-s − 0.498·37-s − 1.24·39-s − 1.31·41-s − 0.380·43-s − 0.000189·45-s − 1.41·47-s + 0.543·49-s + 0.313·51-s + 0.106·53-s + 0.00286·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 0.0212T + 5T^{2} \) |
| 7 | \( 1 - 3.28T + 7T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 + 0.264T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 + 9.67T + 47T^{2} \) |
| 53 | \( 1 - 0.773T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 5.05T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 - 0.0651T + 73T^{2} \) |
| 79 | \( 1 + 3.26T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + 0.480T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.85794312988390788780622222679, −7.24754227868358283015201621524, −6.39639380420686406910315219492, −5.42560087687580769531327920299, −4.76552807776654066646492501116, −4.01809296832115690282294309489, −3.18820910482614344993575187885, −2.09623944187959695830797315987, −1.83028027155593128197821781608, 0,
1.83028027155593128197821781608, 2.09623944187959695830797315987, 3.18820910482614344993575187885, 4.01809296832115690282294309489, 4.76552807776654066646492501116, 5.42560087687580769531327920299, 6.39639380420686406910315219492, 7.24754227868358283015201621524, 7.85794312988390788780622222679
