L(s) = 1 | + 2.48·3-s − 5-s + 7-s + 3.15·9-s + 11-s + 5.83·13-s − 2.48·15-s + 5.44·17-s + 1.35·19-s + 2.48·21-s + 3.19·23-s + 25-s + 0.387·27-s − 3.61·29-s + 5.28·31-s + 2.48·33-s − 35-s − 8.54·37-s + 14.4·39-s − 5.02·41-s − 5.89·43-s − 3.15·45-s + 11.8·47-s + 49-s + 13.5·51-s − 0.231·53-s − 55-s + ⋯ |
L(s) = 1 | + 1.43·3-s − 0.447·5-s + 0.377·7-s + 1.05·9-s + 0.301·11-s + 1.61·13-s − 0.640·15-s + 1.32·17-s + 0.309·19-s + 0.541·21-s + 0.665·23-s + 0.200·25-s + 0.0746·27-s − 0.670·29-s + 0.949·31-s + 0.431·33-s − 0.169·35-s − 1.40·37-s + 2.31·39-s − 0.784·41-s − 0.898·43-s − 0.470·45-s + 1.72·47-s + 0.142·49-s + 1.89·51-s − 0.0318·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.991721743\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.991721743\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 3.19T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 + 5.02T + 41T^{2} \) |
| 43 | \( 1 + 5.89T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 0.231T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 7.22T + 89T^{2} \) |
| 97 | \( 1 + 0.836T + 97T^{2} \) |
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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
−8.193908712852197147649305418364, −7.56068632356121609142392787060, −6.88712676872934444789377855854, −5.93724224493482150111412232080, −5.12812472175213413044072923211, −4.10192453458973379206630969437, −3.46870219936961098021324281856, −3.05897645626141418469610223030, −1.83577095906144636787873474659, −1.07040725550702904979175637306,
1.07040725550702904979175637306, 1.83577095906144636787873474659, 3.05897645626141418469610223030, 3.46870219936961098021324281856, 4.10192453458973379206630969437, 5.12812472175213413044072923211, 5.93724224493482150111412232080, 6.88712676872934444789377855854, 7.56068632356121609142392787060, 8.193908712852197147649305418364