| L(s) = 1 | − 1.93·3-s − 3.21·5-s + 1.58·7-s + 0.761·9-s − 3.51·11-s + 5.00·13-s + 6.23·15-s − 17-s + 4.22·19-s − 3.07·21-s + 5.17·23-s + 5.33·25-s + 4.34·27-s + 2.14·29-s − 3.24·31-s + 6.81·33-s − 5.09·35-s + 5.91·37-s − 9.70·39-s − 4.87·41-s − 3.02·43-s − 2.44·45-s − 3.15·47-s − 4.48·49-s + 1.93·51-s + 3.35·53-s + 11.2·55-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 1.43·5-s + 0.599·7-s + 0.253·9-s − 1.05·11-s + 1.38·13-s + 1.60·15-s − 0.242·17-s + 0.969·19-s − 0.670·21-s + 1.07·23-s + 1.06·25-s + 0.835·27-s + 0.398·29-s − 0.582·31-s + 1.18·33-s − 0.861·35-s + 0.971·37-s − 1.55·39-s − 0.760·41-s − 0.460·43-s − 0.364·45-s − 0.460·47-s − 0.640·49-s + 0.271·51-s + 0.461·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7688437465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7688437465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 - 5.17T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 3.24T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 + 4.87T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 - 3.35T + 53T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 - 2.52T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 1.61T + 73T^{2} \) |
| 79 | \( 1 + 9.71T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 7.17T + 97T^{2} \) |
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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
−7.73714417284348519030245961885, −7.24902821431243302754460497123, −6.39369514332455283007330012938, −5.68950714498460766801081623381, −4.96579455161415831192830022337, −4.53426894251975084849590008228, −3.52748225132538949941962872044, −2.92063575203639079409960782187, −1.40144397801590341500860491091, −0.49931636245972271033588794248,
0.49931636245972271033588794248, 1.40144397801590341500860491091, 2.92063575203639079409960782187, 3.52748225132538949941962872044, 4.53426894251975084849590008228, 4.96579455161415831192830022337, 5.68950714498460766801081623381, 6.39369514332455283007330012938, 7.24902821431243302754460497123, 7.73714417284348519030245961885
