| L(s) = 1 | + 2-s + 3-s + 4-s + 2.74·5-s + 6-s + 8-s + 9-s + 2.74·10-s + 11-s + 12-s + 5.49·13-s + 2.74·15-s + 16-s + 17-s + 18-s − 8.03·19-s + 2.74·20-s + 22-s − 1.29·23-s + 24-s + 2.54·25-s + 5.49·26-s + 27-s + 4.54·29-s + 2.74·30-s + 5.08·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.22·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.868·10-s + 0.301·11-s + 0.288·12-s + 1.52·13-s + 0.709·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.84·19-s + 0.614·20-s + 0.213·22-s − 0.269·23-s + 0.204·24-s + 0.508·25-s + 1.07·26-s + 0.192·27-s + 0.843·29-s + 0.501·30-s + 0.913·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.993088104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.993088104\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.74T + 5T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 8.03T + 19T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 4.54T + 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 + 4.03T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 - 5.49T + 53T^{2} \) |
| 59 | \( 1 + 9.52T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 - 2.54T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 + 0.0872T + 97T^{2} \) |
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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
−8.569053409905107413873061494202, −8.111711200745061641797745684194, −6.81025663878075913355719964462, −6.30298738429695036362198318983, −5.79752033471331040669100413145, −4.70518690229090048788170972575, −3.96877480064951000317686195720, −3.05872133629636959226528848875, −2.12533686355120260634483171459, −1.36731821315587095770166757423,
1.36731821315587095770166757423, 2.12533686355120260634483171459, 3.05872133629636959226528848875, 3.96877480064951000317686195720, 4.70518690229090048788170972575, 5.79752033471331040669100413145, 6.30298738429695036362198318983, 6.81025663878075913355719964462, 8.111711200745061641797745684194, 8.569053409905107413873061494202
