| L(s) = 1 | + 2.54·3-s + 2.87·7-s + 3.46·9-s + 4.99·11-s + 0.149·13-s − 6.23·17-s − 1.90·19-s + 7.32·21-s + 4.35·23-s + 1.19·27-s − 8.93·29-s − 8.99·31-s + 12.7·33-s − 2.56·37-s + 0.379·39-s + 7.99·41-s + 4.54·43-s + 9.68·47-s + 1.28·49-s − 15.8·51-s + 1.52·53-s − 4.85·57-s + 11.4·59-s − 3.55·61-s + 9.98·63-s + 4.08·67-s + 11.0·69-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.08·7-s + 1.15·9-s + 1.50·11-s + 0.0414·13-s − 1.51·17-s − 0.437·19-s + 1.59·21-s + 0.908·23-s + 0.229·27-s − 1.65·29-s − 1.61·31-s + 2.21·33-s − 0.421·37-s + 0.0608·39-s + 1.24·41-s + 0.692·43-s + 1.41·47-s + 0.184·49-s − 2.21·51-s + 0.209·53-s − 0.642·57-s + 1.49·59-s − 0.455·61-s + 1.25·63-s + 0.498·67-s + 1.33·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.994765013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994765013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 - 4.99T + 11T^{2} \) |
| 13 | \( 1 - 0.149T + 13T^{2} \) |
| 17 | \( 1 + 6.23T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 - 4.35T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 37 | \( 1 + 2.56T + 37T^{2} \) |
| 41 | \( 1 - 7.99T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 - 9.68T + 47T^{2} \) |
| 53 | \( 1 - 1.52T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 - 4.08T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 8.96T + 83T^{2} \) |
| 89 | \( 1 + 9.26T + 89T^{2} \) |
| 97 | \( 1 + 3.66T + 97T^{2} \) |
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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
−9.559057462074024632407366483948, −8.922025853648917767473457846565, −8.636910744648432213569931896429, −7.50267654005487894749259981320, −6.94650978289332589074682818951, −5.62847378707304983144190271833, −4.29191986728429138398561444342, −3.80462459054512573600376830957, −2.40219940334343117543772156827, −1.58685374081355208119174043370,
1.58685374081355208119174043370, 2.40219940334343117543772156827, 3.80462459054512573600376830957, 4.29191986728429138398561444342, 5.62847378707304983144190271833, 6.94650978289332589074682818951, 7.50267654005487894749259981320, 8.636910744648432213569931896429, 8.922025853648917767473457846565, 9.559057462074024632407366483948
