| L(s) = 1 | − 1.87·2-s + 1.53·3-s + 2.53·4-s − 2.87·6-s + 1.53·7-s − 2.87·8-s + 1.34·9-s + 3.87·12-s − 2.87·14-s + 2.87·16-s − 2.53·18-s − 1.87·19-s + 2.34·21-s + 0.347·23-s − 4.41·24-s + 25-s + 0.532·27-s + 3.87·28-s − 31-s − 2.53·32-s + 3.41·36-s − 37-s + 3.53·38-s − 41-s − 4.41·42-s − 1.87·43-s − 0.652·46-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 1.53·3-s + 2.53·4-s − 2.87·6-s + 1.53·7-s − 2.87·8-s + 1.34·9-s + 3.87·12-s − 2.87·14-s + 2.87·16-s − 2.53·18-s − 1.87·19-s + 2.34·21-s + 0.347·23-s − 4.41·24-s + 25-s + 0.532·27-s + 3.87·28-s − 31-s − 2.53·32-s + 3.41·36-s − 37-s + 3.53·38-s − 41-s − 4.41·42-s − 1.87·43-s − 0.652·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8389476287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8389476287\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 - T \) |
| good | 2 | \( 1 + 1.87T + T^{2} \) |
| 3 | \( 1 - 1.53T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.53T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.87T + T^{2} \) |
| 23 | \( 1 - 0.347T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + 1.87T + T^{2} \) |
| 47 | \( 1 - 0.347T + T^{2} \) |
| 53 | \( 1 + 1.87T + T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.53T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.00976765839620512240654157833, −8.937592814778045810468873002996, −8.610017094120081651756384364419, −8.121923688161267018332256313561, −7.35753745137102005950734718359, −6.55412167788695120917645013294, −4.90281843828383125309979180412, −3.44254602716621065018635735063, −2.19423643637913643330652866737, −1.66081493320676557575426171011,
1.66081493320676557575426171011, 2.19423643637913643330652866737, 3.44254602716621065018635735063, 4.90281843828383125309979180412, 6.55412167788695120917645013294, 7.35753745137102005950734718359, 8.121923688161267018332256313561, 8.610017094120081651756384364419, 8.937592814778045810468873002996, 10.00976765839620512240654157833
