| L(s) = 1 | + 1.41·3-s − 3.93·5-s + 2.73·7-s − 0.999·9-s − 2.73·11-s − 2.03·13-s − 5.56·15-s + 6.44·17-s + 3.56·19-s + 3.86·21-s + 10.4·25-s − 5.65·27-s − 5.52·29-s + 1.60·31-s − 3.86·33-s − 10.7·35-s + 9.84·37-s − 2.87·39-s − 4.46·41-s − 11.8·43-s + 3.93·45-s + 9.98·47-s + 0.464·49-s + 9.12·51-s + 5.53·53-s + 10.7·55-s + 5.03·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 1.75·5-s + 1.03·7-s − 0.333·9-s − 0.823·11-s − 0.564·13-s − 1.43·15-s + 1.56·17-s + 0.816·19-s + 0.843·21-s + 2.09·25-s − 1.08·27-s − 1.02·29-s + 0.287·31-s − 0.672·33-s − 1.81·35-s + 1.61·37-s − 0.460·39-s − 0.697·41-s − 1.80·43-s + 0.586·45-s + 1.45·47-s + 0.0663·49-s + 1.27·51-s + 0.760·53-s + 1.44·55-s + 0.666·57-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 3.93T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 + 2.03T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 - 9.84T + 37T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 9.98T + 47T^{2} \) |
| 53 | \( 1 - 5.53T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.118T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 - 9.65T + 89T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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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
−7.59179805719991720536646527391, −7.37924377175986520544245279653, −5.95939714041264396673894721988, −5.12565534400499838265946210325, −4.64780598576531845564710443201, −3.65819589647433975302277838043, −3.22658193588104805481176340776, −2.43206559656179885045238841589, −1.20697507360982634155648026888, 0,
1.20697507360982634155648026888, 2.43206559656179885045238841589, 3.22658193588104805481176340776, 3.65819589647433975302277838043, 4.64780598576531845564710443201, 5.12565534400499838265946210325, 5.95939714041264396673894721988, 7.37924377175986520544245279653, 7.59179805719991720536646527391
