| L(s) = 1 | + 0.347·2-s + 3-s − 1.87·4-s + 3.53·5-s + 0.347·6-s − 7-s − 1.34·8-s + 9-s + 1.22·10-s − 3.29·11-s − 1.87·12-s − 0.347·14-s + 3.53·15-s + 3.29·16-s − 5.17·17-s + 0.347·18-s + 6.06·19-s − 6.63·20-s − 21-s − 1.14·22-s + 1.49·23-s − 1.34·24-s + 7.47·25-s + 27-s + 1.87·28-s + 4.73·29-s + 1.22·30-s + ⋯ |
| L(s) = 1 | + 0.245·2-s + 0.577·3-s − 0.939·4-s + 1.57·5-s + 0.141·6-s − 0.377·7-s − 0.476·8-s + 0.333·9-s + 0.387·10-s − 0.992·11-s − 0.542·12-s − 0.0928·14-s + 0.911·15-s + 0.822·16-s − 1.25·17-s + 0.0818·18-s + 1.39·19-s − 1.48·20-s − 0.218·21-s − 0.243·22-s + 0.310·23-s − 0.275·24-s + 1.49·25-s + 0.192·27-s + 0.355·28-s + 0.879·29-s + 0.223·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.586276306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.586276306\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 17 | \( 1 + 5.17T + 17T^{2} \) |
| 19 | \( 1 - 6.06T + 19T^{2} \) |
| 23 | \( 1 - 1.49T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 - 7.90T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 0.162T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 9.22T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 0.0641T + 89T^{2} \) |
| 97 | \( 1 - 0.448T + 97T^{2} \) |
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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
−8.792441609931780698025212593900, −7.978952657585499673493651433392, −7.01589626720278875304683750325, −6.20227158537072456289838948201, −5.38888808477002738332814005301, −4.95118638011673868747817695017, −3.89903890151244480712108289233, −2.86073994069308439431166431864, −2.27858558381304236900852184715, −0.905595143552427927379995013257,
0.905595143552427927379995013257, 2.27858558381304236900852184715, 2.86073994069308439431166431864, 3.89903890151244480712108289233, 4.95118638011673868747817695017, 5.38888808477002738332814005301, 6.20227158537072456289838948201, 7.01589626720278875304683750325, 7.978952657585499673493651433392, 8.792441609931780698025212593900
