| L(s) = 1 | − 2-s − 3-s + 4-s − 2.82·5-s + 6-s + 2.82·7-s − 8-s + 9-s + 2.82·10-s + 5.65·11-s − 12-s + 6·13-s − 2.82·14-s + 2.82·15-s + 16-s + 2.82·17-s − 18-s + 8.48·19-s − 2.82·20-s − 2.82·21-s − 5.65·22-s + 24-s + 3.00·25-s − 6·26-s − 27-s + 2.82·28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.26·5-s + 0.408·6-s + 1.06·7-s − 0.353·8-s + 0.333·9-s + 0.894·10-s + 1.70·11-s − 0.288·12-s + 1.66·13-s − 0.755·14-s + 0.730·15-s + 0.250·16-s + 0.685·17-s − 0.235·18-s + 1.94·19-s − 0.632·20-s − 0.617·21-s − 1.20·22-s + 0.204·24-s + 0.600·25-s − 1.17·26-s − 0.192·27-s + 0.534·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.394008881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394008881\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 - 8.48T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−8.467442331666015688337225470798, −8.068606039548218015100998434360, −7.29618053161278107895417098859, −6.57823208127191553960104266439, −5.77156182878714508192699356966, −4.76948100049499758279424630880, −3.89456860978367748568179861011, −3.27907429098490894701211729020, −1.36267143182134564569821390287, −1.01646682940596376855623981528,
1.01646682940596376855623981528, 1.36267143182134564569821390287, 3.27907429098490894701211729020, 3.89456860978367748568179861011, 4.76948100049499758279424630880, 5.77156182878714508192699356966, 6.57823208127191553960104266439, 7.29618053161278107895417098859, 8.068606039548218015100998434360, 8.467442331666015688337225470798
