| L(s) = 1 | + 1.90·2-s + 0.214·3-s + 1.62·4-s + 5-s + 0.409·6-s − 7-s − 0.710·8-s − 2.95·9-s + 1.90·10-s + 0.349·12-s + 4.56·13-s − 1.90·14-s + 0.214·15-s − 4.60·16-s − 3.25·17-s − 5.62·18-s − 5.02·19-s + 1.62·20-s − 0.214·21-s + 0.292·23-s − 0.152·24-s + 25-s + 8.70·26-s − 1.28·27-s − 1.62·28-s − 1.00·29-s + 0.409·30-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.124·3-s + 0.813·4-s + 0.447·5-s + 0.167·6-s − 0.377·7-s − 0.251·8-s − 0.984·9-s + 0.602·10-s + 0.100·12-s + 1.26·13-s − 0.509·14-s + 0.0555·15-s − 1.15·16-s − 0.788·17-s − 1.32·18-s − 1.15·19-s + 0.363·20-s − 0.0469·21-s + 0.0610·23-s − 0.0311·24-s + 0.200·25-s + 1.70·26-s − 0.246·27-s − 0.307·28-s − 0.186·29-s + 0.0747·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 - 0.214T + 3T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 3.25T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 23 | \( 1 - 0.292T + 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 + 2.69T + 41T^{2} \) |
| 43 | \( 1 + 0.193T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 2.78T + 53T^{2} \) |
| 59 | \( 1 + 0.846T + 59T^{2} \) |
| 61 | \( 1 + 0.963T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 9.39T + 79T^{2} \) |
| 83 | \( 1 + 9.07T + 83T^{2} \) |
| 89 | \( 1 - 4.30T + 89T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−8.120917595324659848144030533681, −6.81858007481845255128226852860, −6.38499800838446616774619399483, −5.72260291513597980949499861474, −5.14002367959133247412397782537, −4.12208264666769934794363095592, −3.54385060295725947639879023941, −2.72399455022574741472333559072, −1.85094081854214560250877149282, 0,
1.85094081854214560250877149282, 2.72399455022574741472333559072, 3.54385060295725947639879023941, 4.12208264666769934794363095592, 5.14002367959133247412397782537, 5.72260291513597980949499861474, 6.38499800838446616774619399483, 6.81858007481845255128226852860, 8.120917595324659848144030533681
