L(s) = 1 | − 3·3-s + 4.31·5-s − 7·7-s + 9·9-s − 41.9·11-s − 26.2·13-s − 12.9·15-s + 57.8·17-s + 2.16·19-s + 21·21-s − 200.·23-s − 106.·25-s − 27·27-s + 121.·29-s + 279.·31-s + 125.·33-s − 30.1·35-s − 124.·37-s + 78.8·39-s + 363.·41-s − 253.·43-s + 38.8·45-s + 84.6·47-s + 49·49-s − 173.·51-s − 507.·53-s − 181.·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.385·5-s − 0.377·7-s + 0.333·9-s − 1.15·11-s − 0.560·13-s − 0.222·15-s + 0.825·17-s + 0.0261·19-s + 0.218·21-s − 1.82·23-s − 0.851·25-s − 0.192·27-s + 0.776·29-s + 1.62·31-s + 0.664·33-s − 0.145·35-s − 0.552·37-s + 0.323·39-s + 1.38·41-s − 0.898·43-s + 0.128·45-s + 0.262·47-s + 0.142·49-s − 0.476·51-s − 1.31·53-s − 0.443·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.162156714\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.162156714\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
good | 5 | \( 1 - 4.31T + 125T^{2} \) |
| 11 | \( 1 + 41.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.16T + 6.85e3T^{2} \) |
| 23 | \( 1 + 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 363.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 253.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 84.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 507.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 529.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 429.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 645.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 974.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 302.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.14e3T + 9.12e5T^{2} \) |
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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
−9.617897976036015176944339167359, −8.183340382916495792286273973937, −7.75129048586197745267214480398, −6.60897190114826605104654667455, −5.89838162610113166255076279193, −5.17836799348553965876181263115, −4.22604246466596444307521324181, −2.98692777541772668901223241756, −1.99198103166180871729494874080, −0.53649615068815398218941122846,
0.53649615068815398218941122846, 1.99198103166180871729494874080, 2.98692777541772668901223241756, 4.22604246466596444307521324181, 5.17836799348553965876181263115, 5.89838162610113166255076279193, 6.60897190114826605104654667455, 7.75129048586197745267214480398, 8.183340382916495792286273973937, 9.617897976036015176944339167359