| L(s) = 1 | + 2.40·2-s + 2.03·3-s + 3.80·4-s + 4.91·6-s + 0.999·7-s + 4.33·8-s + 1.15·9-s + 3.23·11-s + 7.75·12-s − 1.80·13-s + 2.40·14-s + 2.84·16-s − 3.81·17-s + 2.79·18-s + 8.53·19-s + 2.03·21-s + 7.78·22-s + 6.96·23-s + 8.84·24-s − 4.34·26-s − 3.75·27-s + 3.79·28-s − 1.36·29-s − 2.86·31-s − 1.81·32-s + 6.58·33-s − 9.17·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.17·3-s + 1.90·4-s + 2.00·6-s + 0.377·7-s + 1.53·8-s + 0.386·9-s + 0.974·11-s + 2.23·12-s − 0.499·13-s + 0.643·14-s + 0.712·16-s − 0.924·17-s + 0.657·18-s + 1.95·19-s + 0.444·21-s + 1.65·22-s + 1.45·23-s + 1.80·24-s − 0.851·26-s − 0.722·27-s + 0.718·28-s − 0.253·29-s − 0.514·31-s − 0.321·32-s + 1.14·33-s − 1.57·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.570574028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.570574028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 - 2.03T + 3T^{2} \) |
| 7 | \( 1 - 0.999T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 - 8.53T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 - 8.29T + 37T^{2} \) |
| 41 | \( 1 - 0.597T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 0.0569T + 71T^{2} \) |
| 73 | \( 1 - 2.71T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 8.65T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 9.68T + 97T^{2} \) |
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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
−8.404653351968314759357688673942, −7.58057612832540798430986612575, −6.93814980956133548047549672664, −6.24188460668715528283592817709, −5.13661943813360694759000120994, −4.78092397591326791580243501890, −3.68374786682481704492003501772, −3.25654257674552526712536223441, −2.45396233593969006800548382171, −1.50035613893272655430173198023,
1.50035613893272655430173198023, 2.45396233593969006800548382171, 3.25654257674552526712536223441, 3.68374786682481704492003501772, 4.78092397591326791580243501890, 5.13661943813360694759000120994, 6.24188460668715528283592817709, 6.93814980956133548047549672664, 7.58057612832540798430986612575, 8.404653351968314759357688673942
