| L(s) = 1 | + 3-s + 4.08·5-s + 2.57·7-s + 9-s + 4.61·11-s − 4.79·13-s + 4.08·15-s − 1.83·17-s + 4.89·19-s + 2.57·21-s + 23-s + 11.7·25-s + 27-s − 29-s + 8.03·31-s + 4.61·33-s + 10.5·35-s − 8.43·37-s − 4.79·39-s + 8.64·41-s − 7.46·43-s + 4.08·45-s − 5.52·47-s − 0.366·49-s − 1.83·51-s − 7.81·53-s + 18.8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.82·5-s + 0.973·7-s + 0.333·9-s + 1.39·11-s − 1.33·13-s + 1.05·15-s − 0.444·17-s + 1.12·19-s + 0.562·21-s + 0.208·23-s + 2.34·25-s + 0.192·27-s − 0.185·29-s + 1.44·31-s + 0.802·33-s + 1.78·35-s − 1.38·37-s − 0.768·39-s + 1.35·41-s − 1.13·43-s + 0.609·45-s − 0.806·47-s − 0.0523·49-s − 0.256·51-s − 1.07·53-s + 2.54·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.814340552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.814340552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 4.08T + 5T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 31 | \( 1 - 8.03T + 31T^{2} \) |
| 37 | \( 1 + 8.43T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 + 7.46T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 + 7.81T + 53T^{2} \) |
| 59 | \( 1 - 7.57T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 8.30T + 67T^{2} \) |
| 71 | \( 1 + 0.0870T + 71T^{2} \) |
| 73 | \( 1 - 9.17T + 73T^{2} \) |
| 79 | \( 1 + 8.38T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 6.00T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87905067566021583287326284538, −6.99506400964558501249850294469, −6.57864900416905891940992811982, −5.71075278389378831813693057728, −4.96381560764056595775648459417, −4.54709495802633873469249965513, −3.31314168616254975029579341651, −2.50028287319415765291363856510, −1.77301526862498186376390551942, −1.20269040816193260036434779832,
1.20269040816193260036434779832, 1.77301526862498186376390551942, 2.50028287319415765291363856510, 3.31314168616254975029579341651, 4.54709495802633873469249965513, 4.96381560764056595775648459417, 5.71075278389378831813693057728, 6.57864900416905891940992811982, 6.99506400964558501249850294469, 7.87905067566021583287326284538
