| L(s) = 1 | + 2-s + 4-s + 4.12·5-s − 4.21·7-s + 8-s + 4.12·10-s + 11-s − 2.21·13-s − 4.21·14-s + 16-s + 3.45·17-s − 19-s + 4.12·20-s + 22-s + 5.45·23-s + 12.0·25-s − 2.21·26-s − 4.21·28-s − 5.57·29-s + 7.00·31-s + 32-s + 3.45·34-s − 17.3·35-s − 2.90·37-s − 38-s + 4.12·40-s + 11.9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.84·5-s − 1.59·7-s + 0.353·8-s + 1.30·10-s + 0.301·11-s − 0.614·13-s − 1.12·14-s + 0.250·16-s + 0.837·17-s − 0.229·19-s + 0.922·20-s + 0.213·22-s + 1.13·23-s + 2.40·25-s − 0.434·26-s − 0.796·28-s − 1.03·29-s + 1.25·31-s + 0.176·32-s + 0.592·34-s − 2.93·35-s − 0.478·37-s − 0.162·38-s + 0.652·40-s + 1.86·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.813527915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.813527915\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 4.12T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 - 7.00T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 8.90T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 - 6.80T + 71T^{2} \) |
| 73 | \( 1 + 1.45T + 73T^{2} \) |
| 79 | \( 1 + 9.15T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−8.777330827461094416869310682549, −7.42287122894200064408034787332, −6.73205219702496126636342677607, −6.19833382264259948326580291927, −5.61780377378520884971941482264, −4.94620382366068598508525214525, −3.76209089006245071027610009514, −2.87028323279788616978177319180, −2.32825021569753419400491368980, −1.05988812158155640401461105704,
1.05988812158155640401461105704, 2.32825021569753419400491368980, 2.87028323279788616978177319180, 3.76209089006245071027610009514, 4.94620382366068598508525214525, 5.61780377378520884971941482264, 6.19833382264259948326580291927, 6.73205219702496126636342677607, 7.42287122894200064408034787332, 8.777330827461094416869310682549
