| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 3.46·13-s + 14-s + 15-s + 16-s − 3.46·17-s − 18-s + 20-s − 21-s − 22-s − 1.46·23-s − 24-s + 25-s + 3.46·26-s + 27-s − 28-s + 8.92·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.960·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s − 0.840·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.305·23-s − 0.204·24-s + 0.200·25-s + 0.679·26-s + 0.192·27-s − 0.188·28-s + 1.65·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609757071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609757071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.46T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 5.46T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.991187076116031992991246591110, −8.380694225009657755698764156911, −7.58668995396088903980732021247, −6.70960634929088182198961383864, −6.23634273714901971750450507631, −4.98104918234492355914653584014, −4.10544537411848441830162153941, −2.79793511663247812338419173423, −2.29468685460817520592942856162, −0.888788568762048489544398883449,
0.888788568762048489544398883449, 2.29468685460817520592942856162, 2.79793511663247812338419173423, 4.10544537411848441830162153941, 4.98104918234492355914653584014, 6.23634273714901971750450507631, 6.70960634929088182198961383864, 7.58668995396088903980732021247, 8.380694225009657755698764156911, 8.991187076116031992991246591110
