| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 6·13-s − 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s − 4·22-s + 8·23-s + 24-s − 25-s + 6·26-s − 27-s + 2·29-s + 2·30-s + 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.852·22-s + 1.66·23-s + 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9851349914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9851349914\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−11.53741449340409014874451507343, −10.26722383012066951313519422841, −9.707131427984874107304014072925, −8.997351406334560750563905507145, −7.54748358826902146820019930668, −6.76473781332737149602915444418, −5.76646656864847315102908194383, −4.69366950891260328755548006577, −2.80942620041714391710386178009, −1.24206810304200841915782160563,
1.24206810304200841915782160563, 2.80942620041714391710386178009, 4.69366950891260328755548006577, 5.76646656864847315102908194383, 6.76473781332737149602915444418, 7.54748358826902146820019930668, 8.997351406334560750563905507145, 9.707131427984874107304014072925, 10.26722383012066951313519422841, 11.53741449340409014874451507343
