| L(s) = 1 | − 2.76·2-s + 1.30·3-s + 5.64·4-s − 2.58·5-s − 3.61·6-s − 0.870·7-s − 10.0·8-s − 1.28·9-s + 7.14·10-s − 1.36·11-s + 7.38·12-s − 1.36·13-s + 2.40·14-s − 3.37·15-s + 16.5·16-s + 6.73·17-s + 3.56·18-s + 0.887·19-s − 14.5·20-s − 1.13·21-s + 3.78·22-s − 8.40·23-s − 13.1·24-s + 1.67·25-s + 3.78·26-s − 5.61·27-s − 4.91·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.755·3-s + 2.82·4-s − 1.15·5-s − 1.47·6-s − 0.329·7-s − 3.56·8-s − 0.429·9-s + 2.25·10-s − 0.412·11-s + 2.13·12-s − 0.379·13-s + 0.643·14-s − 0.872·15-s + 4.14·16-s + 1.63·17-s + 0.839·18-s + 0.203·19-s − 3.26·20-s − 0.248·21-s + 0.806·22-s − 1.75·23-s − 2.69·24-s + 0.334·25-s + 0.742·26-s − 1.07·27-s − 0.929·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3126860631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3126860631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 7 | \( 1 + 0.870T + 7T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 0.887T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 + 5.53T + 29T^{2} \) |
| 31 | \( 1 - 9.82T + 31T^{2} \) |
| 37 | \( 1 + 1.35T + 37T^{2} \) |
| 41 | \( 1 + 9.71T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 - 4.14T + 47T^{2} \) |
| 59 | \( 1 + 9.95T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.99T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.764T + 89T^{2} \) |
| 97 | \( 1 + 7.44T + 97T^{2} \) |
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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
−8.026929705738912595051915178296, −7.80395820950608174474232807748, −6.82465177143673983909825166417, −6.12214790237797067431178195371, −5.27232843482646921019013300851, −3.75105782721128248416203846829, −3.24476794606023004590913769103, −2.52077619865721457775603253603, −1.59079487780147336623326575769, −0.34958757305562506050371927519,
0.34958757305562506050371927519, 1.59079487780147336623326575769, 2.52077619865721457775603253603, 3.24476794606023004590913769103, 3.75105782721128248416203846829, 5.27232843482646921019013300851, 6.12214790237797067431178195371, 6.82465177143673983909825166417, 7.80395820950608174474232807748, 8.026929705738912595051915178296
