| L(s) = 1 | − 2.16·3-s + 3.18·5-s + 1.65·7-s + 1.69·9-s + 1.87·11-s − 1.01·13-s − 6.90·15-s + 17-s − 1.11·19-s − 3.59·21-s + 1.18·23-s + 5.15·25-s + 2.83·27-s − 7.37·29-s + 10.6·31-s − 4.06·33-s + 5.28·35-s + 2.75·37-s + 2.19·39-s − 7.46·41-s + 3.91·43-s + 5.39·45-s − 10.3·47-s − 4.25·49-s − 2.16·51-s + 2.85·53-s + 5.97·55-s + ⋯ |
| L(s) = 1 | − 1.25·3-s + 1.42·5-s + 0.626·7-s + 0.564·9-s + 0.565·11-s − 0.281·13-s − 1.78·15-s + 0.242·17-s − 0.255·19-s − 0.783·21-s + 0.246·23-s + 1.03·25-s + 0.544·27-s − 1.36·29-s + 1.91·31-s − 0.707·33-s + 0.892·35-s + 0.452·37-s + 0.351·39-s − 1.16·41-s + 0.596·43-s + 0.804·45-s − 1.51·47-s − 0.607·49-s − 0.303·51-s + 0.392·53-s + 0.806·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.925285821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.925285821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 + 1.01T + 13T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 + 7.37T + 29T^{2} \) |
| 31 | \( 1 - 10.6T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 + 7.46T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 61 | \( 1 - 0.465T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 2.88T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 7.92T + 89T^{2} \) |
| 97 | \( 1 + 8.06T + 97T^{2} \) |
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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
−7.82520386530030833085504360239, −6.68128583380106055928557224497, −6.46393566285050540715812575749, −5.74719202817839843173999928171, −5.09521723904528137753347600316, −4.72863821468759231641273228008, −3.56993857061247311990627408050, −2.41110824648047859655455363982, −1.65636375879830153850428009865, −0.76588407749767762750995678097,
0.76588407749767762750995678097, 1.65636375879830153850428009865, 2.41110824648047859655455363982, 3.56993857061247311990627408050, 4.72863821468759231641273228008, 5.09521723904528137753347600316, 5.74719202817839843173999928171, 6.46393566285050540715812575749, 6.68128583380106055928557224497, 7.82520386530030833085504360239
