| L(s) = 1 | − 2.14·2-s − 1.72·3-s + 2.59·4-s + 3.54·5-s + 3.69·6-s + 2.57·7-s − 1.28·8-s − 0.0251·9-s − 7.60·10-s − 11-s − 4.48·12-s − 5.51·14-s − 6.11·15-s − 2.44·16-s + 4.72·17-s + 0.0539·18-s − 4.99·19-s + 9.21·20-s − 4.43·21-s + 2.14·22-s + 4.05·23-s + 2.21·24-s + 7.57·25-s + 5.21·27-s + 6.67·28-s + 7.00·29-s + 13.1·30-s + ⋯ |
| L(s) = 1 | − 1.51·2-s − 0.995·3-s + 1.29·4-s + 1.58·5-s + 1.50·6-s + 0.971·7-s − 0.453·8-s − 0.00838·9-s − 2.40·10-s − 0.301·11-s − 1.29·12-s − 1.47·14-s − 1.57·15-s − 0.611·16-s + 1.14·17-s + 0.0127·18-s − 1.14·19-s + 2.06·20-s − 0.967·21-s + 0.457·22-s + 0.846·23-s + 0.451·24-s + 1.51·25-s + 1.00·27-s + 1.26·28-s + 1.30·29-s + 2.39·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8553999236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8553999236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 3 | \( 1 + 1.72T + 3T^{2} \) |
| 5 | \( 1 - 3.54T + 5T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 + 1.93T + 41T^{2} \) |
| 43 | \( 1 + 6.58T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 4.22T + 71T^{2} \) |
| 73 | \( 1 - 0.634T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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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
−9.428519695407255464280794595544, −8.325335371240927366410097591110, −8.117931613040154314409262196675, −6.65853293259390640850151366562, −6.38107145678802741761903301451, −5.24237762178854479633310091635, −4.82800039529786315699576649470, −2.73988862825726001021023574373, −1.73961371900895533738079490623, −0.879154769061083942187337210308,
0.879154769061083942187337210308, 1.73961371900895533738079490623, 2.73988862825726001021023574373, 4.82800039529786315699576649470, 5.24237762178854479633310091635, 6.38107145678802741761903301451, 6.65853293259390640850151366562, 8.117931613040154314409262196675, 8.325335371240927366410097591110, 9.428519695407255464280794595544
