| L(s) = 1 | + 0.763·7-s + 5.70·23-s + 6·29-s + 4.47·41-s + 11.2·43-s − 13.7·47-s − 6.41·49-s − 13.4·61-s − 8.18·67-s + 17.7·83-s − 6·89-s + 18·101-s + 20.1·103-s + 6.29·107-s + 13.4·109-s + ⋯ |
| L(s) = 1 | + 0.288·7-s + 1.19·23-s + 1.11·29-s + 0.698·41-s + 1.71·43-s − 1.99·47-s − 0.916·49-s − 1.71·61-s − 0.999·67-s + 1.94·83-s − 0.635·89-s + 1.79·101-s + 1.98·103-s + 0.608·107-s + 1.28·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.127884251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127884251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.763T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 13.7T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8.18T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.83635595286014022258095023805, −7.33821538064769685917881881518, −6.45660213987737243046529437775, −5.94103262162492931038179789181, −4.87903609808944360686114794223, −4.58767631266704864393775510878, −3.44818114533106543883266761275, −2.79107918921604666135985678376, −1.75800995098082718900107585275, −0.75755906389362835296778853913,
0.75755906389362835296778853913, 1.75800995098082718900107585275, 2.79107918921604666135985678376, 3.44818114533106543883266761275, 4.58767631266704864393775510878, 4.87903609808944360686114794223, 5.94103262162492931038179789181, 6.45660213987737243046529437775, 7.33821538064769685917881881518, 7.83635595286014022258095023805
