| L(s) = 1 | + 3-s + 4.92·7-s + 9-s − 1.38·11-s − 13-s − 0.195·17-s + 4.92·21-s + 2.19·23-s + 27-s + 7.49·29-s − 1.38·33-s − 6.05·37-s − 39-s − 2.11·41-s − 3.49·43-s − 2.31·47-s + 17.2·49-s − 0.195·51-s + 6.05·53-s + 9.43·59-s − 3.69·61-s + 4.92·63-s + 12.6·67-s + 2.19·69-s + 13.9·71-s − 4.73·73-s − 6.81·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.86·7-s + 0.333·9-s − 0.417·11-s − 0.277·13-s − 0.0473·17-s + 1.07·21-s + 0.457·23-s + 0.192·27-s + 1.39·29-s − 0.240·33-s − 0.994·37-s − 0.160·39-s − 0.330·41-s − 0.533·43-s − 0.337·47-s + 2.46·49-s − 0.0273·51-s + 0.831·53-s + 1.22·59-s − 0.473·61-s + 0.620·63-s + 1.54·67-s + 0.264·69-s + 1.65·71-s − 0.553·73-s − 0.776·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.395410097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.395410097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 17 | \( 1 + 0.195T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.05T + 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 + 3.49T + 43T^{2} \) |
| 47 | \( 1 + 2.31T + 47T^{2} \) |
| 53 | \( 1 - 6.05T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 4.73T + 73T^{2} \) |
| 79 | \( 1 + 8.07T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 3.02T + 89T^{2} \) |
| 97 | \( 1 + 6.01T + 97T^{2} \) |
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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
−8.186596539982549977663567060070, −7.24012532903157890762542665052, −6.71341870777901671302674281551, −5.50116811892375664339324999793, −4.99675035837101901967681012314, −4.42897146406942717021275356424, −3.52014309589358310634831340341, −2.52370258772697067496418962887, −1.87755573750994102547755197948, −0.936028014058307779117766079293,
0.936028014058307779117766079293, 1.87755573750994102547755197948, 2.52370258772697067496418962887, 3.52014309589358310634831340341, 4.42897146406942717021275356424, 4.99675035837101901967681012314, 5.50116811892375664339324999793, 6.71341870777901671302674281551, 7.24012532903157890762542665052, 8.186596539982549977663567060070
