| L(s) = 1 | + 1.45·3-s + 3.79·5-s − 3.95·7-s − 0.891·9-s − 2.06·11-s + 3.18·13-s + 5.50·15-s + 6.58·17-s − 5.74·21-s + 7.10·23-s + 9.37·25-s − 5.65·27-s − 3.19·29-s + 3.24·31-s − 3.00·33-s − 15.0·35-s + 0.364·37-s + 4.62·39-s + 11.3·41-s − 4.39·43-s − 3.37·45-s − 4.46·47-s + 8.66·49-s + 9.55·51-s + 4.70·53-s − 7.83·55-s + 10.3·59-s + ⋯ |
| L(s) = 1 | + 0.838·3-s + 1.69·5-s − 1.49·7-s − 0.297·9-s − 0.623·11-s + 0.883·13-s + 1.42·15-s + 1.59·17-s − 1.25·21-s + 1.48·23-s + 1.87·25-s − 1.08·27-s − 0.592·29-s + 0.583·31-s − 0.522·33-s − 2.53·35-s + 0.0598·37-s + 0.740·39-s + 1.77·41-s − 0.670·43-s − 0.503·45-s − 0.652·47-s + 1.23·49-s + 1.33·51-s + 0.646·53-s − 1.05·55-s + 1.34·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.979507159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.979507159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 - 3.79T + 5T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 - 3.18T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 23 | \( 1 - 7.10T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 0.364T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 + 4.39T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 + 4.72T + 79T^{2} \) |
| 83 | \( 1 + 4.04T + 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.047343081828218062699574480258, −8.161162011698685847456631924425, −7.22846791428792941920082324939, −6.31605726287311378143774906017, −5.81875115343979756452079793950, −5.17481804356893459967564937617, −3.58714888727795413600857877186, −3.00960168805064195539026222215, −2.35540086961771203370108501042, −1.06132744313796677867147785709,
1.06132744313796677867147785709, 2.35540086961771203370108501042, 3.00960168805064195539026222215, 3.58714888727795413600857877186, 5.17481804356893459967564937617, 5.81875115343979756452079793950, 6.31605726287311378143774906017, 7.22846791428792941920082324939, 8.161162011698685847456631924425, 9.047343081828218062699574480258
