| L(s) = 1 | + 2-s + 3-s + 4-s + 0.566·5-s + 6-s + 2.05·7-s + 8-s + 9-s + 0.566·10-s + 5.08·11-s + 12-s + 1.56·13-s + 2.05·14-s + 0.566·15-s + 16-s − 0.517·17-s + 18-s − 2.11·19-s + 0.566·20-s + 2.05·21-s + 5.08·22-s + 23-s + 24-s − 4.67·25-s + 1.56·26-s + 27-s + 2.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.253·5-s + 0.408·6-s + 0.777·7-s + 0.353·8-s + 0.333·9-s + 0.179·10-s + 1.53·11-s + 0.288·12-s + 0.433·13-s + 0.549·14-s + 0.146·15-s + 0.250·16-s − 0.125·17-s + 0.235·18-s − 0.486·19-s + 0.126·20-s + 0.449·21-s + 1.08·22-s + 0.208·23-s + 0.204·24-s − 0.935·25-s + 0.306·26-s + 0.192·27-s + 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.969252514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.969252514\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 0.566T + 5T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 - 5.08T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 + 0.517T + 17T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 31 | \( 1 - 8.44T + 31T^{2} \) |
| 37 | \( 1 + 6.69T + 37T^{2} \) |
| 41 | \( 1 - 0.665T + 41T^{2} \) |
| 43 | \( 1 - 8.41T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + 9.82T + 53T^{2} \) |
| 59 | \( 1 + 1.56T + 59T^{2} \) |
| 61 | \( 1 + 1.95T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 - 4.20T + 71T^{2} \) |
| 73 | \( 1 - 3.31T + 73T^{2} \) |
| 79 | \( 1 - 2.03T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 5.49T + 97T^{2} \) |
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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
−8.334820948740492039684789962784, −7.80877106764465711547271672060, −6.74902059986344580560449339209, −6.34919042596090697953913801523, −5.41003241869187469501998246930, −4.45637800113426075774661037368, −3.98303175230720794312701848991, −3.07087302709559797909096216197, −1.98812281786323207168452360715, −1.29401530730114921092194384508,
1.29401530730114921092194384508, 1.98812281786323207168452360715, 3.07087302709559797909096216197, 3.98303175230720794312701848991, 4.45637800113426075774661037368, 5.41003241869187469501998246930, 6.34919042596090697953913801523, 6.74902059986344580560449339209, 7.80877106764465711547271672060, 8.334820948740492039684789962784
