| L(s) = 1 | − 2.27·2-s + 1.50·3-s + 3.17·4-s + 5-s − 3.42·6-s + 2.50·7-s − 2.68·8-s − 0.736·9-s − 2.27·10-s − 5.24·11-s + 4.78·12-s − 3.80·13-s − 5.69·14-s + 1.50·15-s − 0.255·16-s − 4.81·17-s + 1.67·18-s − 6.96·19-s + 3.17·20-s + 3.76·21-s + 11.9·22-s + 7.29·23-s − 4.03·24-s + 25-s + 8.65·26-s − 5.62·27-s + 7.95·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 0.868·3-s + 1.58·4-s + 0.447·5-s − 1.39·6-s + 0.946·7-s − 0.947·8-s − 0.245·9-s − 0.719·10-s − 1.58·11-s + 1.38·12-s − 1.05·13-s − 1.52·14-s + 0.388·15-s − 0.0639·16-s − 1.16·17-s + 0.395·18-s − 1.59·19-s + 0.710·20-s + 0.822·21-s + 2.54·22-s + 1.52·23-s − 0.823·24-s + 0.200·25-s + 1.69·26-s − 1.08·27-s + 1.50·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8800339956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8800339956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 1.50T + 3T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 + 4.81T + 17T^{2} \) |
| 19 | \( 1 + 6.96T + 19T^{2} \) |
| 23 | \( 1 - 7.29T + 23T^{2} \) |
| 29 | \( 1 - 4.20T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 + 2.66T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 6.25T + 43T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 4.75T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 4.86T + 71T^{2} \) |
| 73 | \( 1 + 6.55T + 73T^{2} \) |
| 79 | \( 1 - 9.02T + 79T^{2} \) |
| 83 | \( 1 + 3.79T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 2.97T + 97T^{2} \) |
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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
−8.140940209358959914517174019440, −7.36353916633500500478296758968, −6.96183964625596438467060546384, −5.91077453569768675404836309092, −4.96864855368071875264859875099, −4.43363477295043490062864183973, −2.84907268126330312397839384381, −2.38360777696578747081228637150, −1.92358554749636751336045249237, −0.53446531757891935050190492573,
0.53446531757891935050190492573, 1.92358554749636751336045249237, 2.38360777696578747081228637150, 2.84907268126330312397839384381, 4.43363477295043490062864183973, 4.96864855368071875264859875099, 5.91077453569768675404836309092, 6.96183964625596438467060546384, 7.36353916633500500478296758968, 8.140940209358959914517174019440
