| L(s) = 1 | + 1.94·2-s − 1.53·3-s + 1.77·4-s + 1.30·5-s − 2.98·6-s − 3.17·7-s − 0.439·8-s − 0.635·9-s + 2.53·10-s + 2.42·11-s − 2.72·12-s − 5.43·13-s − 6.16·14-s − 2.01·15-s − 4.40·16-s + 17-s − 1.23·18-s + 2.44·19-s + 2.31·20-s + 4.87·21-s + 4.71·22-s − 8.76·23-s + 0.676·24-s − 3.29·25-s − 10.5·26-s + 5.59·27-s − 5.62·28-s + ⋯ |
| L(s) = 1 | + 1.37·2-s − 0.887·3-s + 0.886·4-s + 0.584·5-s − 1.21·6-s − 1.19·7-s − 0.155·8-s − 0.211·9-s + 0.803·10-s + 0.731·11-s − 0.787·12-s − 1.50·13-s − 1.64·14-s − 0.519·15-s − 1.10·16-s + 0.242·17-s − 0.290·18-s + 0.559·19-s + 0.518·20-s + 1.06·21-s + 1.00·22-s − 1.82·23-s + 0.138·24-s − 0.658·25-s − 2.07·26-s + 1.07·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 1.94T + 2T^{2} \) |
| 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 + 5.43T + 13T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 8.76T + 23T^{2} \) |
| 29 | \( 1 - 2.08T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 + 8.01T + 37T^{2} \) |
| 41 | \( 1 - 0.176T + 41T^{2} \) |
| 43 | \( 1 - 9.91T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−9.735186000223258096584538814779, −8.999233787996629625150939785969, −7.41711424430250055414149696887, −6.51640540636937522784965994875, −5.90370016601871326847436394655, −5.38509856800935319440351044085, −4.33146103861011677747889311631, −3.35282375734218953574086952995, −2.29529816020235326705233657726, 0,
2.29529816020235326705233657726, 3.35282375734218953574086952995, 4.33146103861011677747889311631, 5.38509856800935319440351044085, 5.90370016601871326847436394655, 6.51640540636937522784965994875, 7.41711424430250055414149696887, 8.999233787996629625150939785969, 9.735186000223258096584538814779
