| L(s) = 1 | − 2·5-s − 7-s + 2·11-s − 3.12·13-s − 3.12·17-s − 19-s − 7.12·23-s − 25-s + 9.12·29-s − 1.12·31-s + 2·35-s − 0.876·37-s − 8.24·41-s + 4·43-s + 49-s + 5.12·53-s − 4·55-s + 6.24·59-s − 2·61-s + 6.24·65-s − 14.2·67-s − 9.36·71-s − 10·73-s − 2·77-s + 13.1·79-s − 9.12·83-s + 6.24·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s − 0.866·13-s − 0.757·17-s − 0.229·19-s − 1.48·23-s − 0.200·25-s + 1.69·29-s − 0.201·31-s + 0.338·35-s − 0.144·37-s − 1.28·41-s + 0.609·43-s + 0.142·49-s + 0.703·53-s − 0.539·55-s + 0.813·59-s − 0.256·61-s + 0.774·65-s − 1.74·67-s − 1.11·71-s − 1.17·73-s − 0.227·77-s + 1.47·79-s − 1.00·83-s + 0.677·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8519307564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8519307564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 - 9.12T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 + 0.876T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 9.36T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + 0.246T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−7.60358830764236590411092016299, −7.07474396074337534150988546293, −6.39649675499337916231273539640, −5.75058348493841252944320472383, −4.66293939515861899167726718694, −4.26791466512881524167539492356, −3.51297941035596890293402275489, −2.66144195653829354079566755847, −1.76980991153540768376863713602, −0.42607731870913130939555465244,
0.42607731870913130939555465244, 1.76980991153540768376863713602, 2.66144195653829354079566755847, 3.51297941035596890293402275489, 4.26791466512881524167539492356, 4.66293939515861899167726718694, 5.75058348493841252944320472383, 6.39649675499337916231273539640, 7.07474396074337534150988546293, 7.60358830764236590411092016299
