| L(s) = 1 | − 1.56·2-s − 3-s + 0.460·4-s + 4.07·5-s + 1.56·6-s + 2.41·8-s + 9-s − 6.38·10-s − 4.60·11-s − 0.460·12-s + 13-s − 4.07·15-s − 4.70·16-s + 3.53·17-s − 1.56·18-s + 5.76·19-s + 1.87·20-s + 7.21·22-s − 4.38·23-s − 2.41·24-s + 11.5·25-s − 1.56·26-s − 27-s + 4.06·29-s + 6.38·30-s − 3.28·31-s + 2.55·32-s + ⋯ |
| L(s) = 1 | − 1.10·2-s − 0.577·3-s + 0.230·4-s + 1.82·5-s + 0.640·6-s + 0.853·8-s + 0.333·9-s − 2.01·10-s − 1.38·11-s − 0.132·12-s + 0.277·13-s − 1.05·15-s − 1.17·16-s + 0.858·17-s − 0.369·18-s + 1.32·19-s + 0.418·20-s + 1.53·22-s − 0.914·23-s − 0.493·24-s + 2.31·25-s − 0.307·26-s − 0.192·27-s + 0.755·29-s + 1.16·30-s − 0.589·31-s + 0.451·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.043657233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.043657233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 - 7.46T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 + 0.567T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 1.93T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 - 8.69T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−9.490872312845809805054464199569, −8.541282289432579679238285902966, −7.74188896514947865201424193461, −6.99028130490738528363212508621, −5.80917823059098234575407599525, −5.52832328609502765099341688880, −4.58140498403958943870895002196, −2.90402612906341516457952026594, −1.85795953350560862690794246697, −0.878325508263627452232577423425,
0.878325508263627452232577423425, 1.85795953350560862690794246697, 2.90402612906341516457952026594, 4.58140498403958943870895002196, 5.52832328609502765099341688880, 5.80917823059098234575407599525, 6.99028130490738528363212508621, 7.74188896514947865201424193461, 8.541282289432579679238285902966, 9.490872312845809805054464199569
