| L(s) = 1 | − 2.34·2-s − 1.89·3-s + 3.48·4-s − 4.06·5-s + 4.43·6-s + 3.16·7-s − 3.47·8-s + 0.589·9-s + 9.51·10-s + 5.59·11-s − 6.60·12-s − 4.59·13-s − 7.40·14-s + 7.70·15-s + 1.17·16-s + 4.80·17-s − 1.38·18-s − 5.33·19-s − 14.1·20-s − 5.99·21-s − 13.1·22-s − 4.15·23-s + 6.58·24-s + 11.5·25-s + 10.7·26-s + 4.56·27-s + 11.0·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 1.09·3-s + 1.74·4-s − 1.81·5-s + 1.81·6-s + 1.19·7-s − 1.22·8-s + 0.196·9-s + 3.00·10-s + 1.68·11-s − 1.90·12-s − 1.27·13-s − 1.97·14-s + 1.98·15-s + 0.293·16-s + 1.16·17-s − 0.325·18-s − 1.22·19-s − 3.16·20-s − 1.30·21-s − 2.79·22-s − 0.866·23-s + 1.34·24-s + 2.30·25-s + 2.10·26-s + 0.878·27-s + 2.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3748535336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3748535336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 + 1.89T + 3T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 23 | \( 1 + 4.15T + 23T^{2} \) |
| 29 | \( 1 - 9.77T + 29T^{2} \) |
| 31 | \( 1 + 0.132T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + 3.68T + 41T^{2} \) |
| 43 | \( 1 - 9.30T + 43T^{2} \) |
| 47 | \( 1 - 7.95T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + 2.16T + 67T^{2} \) |
| 71 | \( 1 + 2.27T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 7.58T + 83T^{2} \) |
| 89 | \( 1 + 6.84T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.457061238161088294091993734496, −7.85560542091909629640949304243, −7.15449375418704411712605775288, −6.70151809657076781550669512805, −5.63122062871358027571745483634, −4.47005826229298741851317147156, −4.16376778726400635093909438898, −2.63115624485588455232929400026, −1.27663083297040030161377828271, −0.55246530780152525899365969338,
0.55246530780152525899365969338, 1.27663083297040030161377828271, 2.63115624485588455232929400026, 4.16376778726400635093909438898, 4.47005826229298741851317147156, 5.63122062871358027571745483634, 6.70151809657076781550669512805, 7.15449375418704411712605775288, 7.85560542091909629640949304243, 8.457061238161088294091993734496
