L(s) = 1 | + 0.972·3-s + 0.716·5-s − 2.42·7-s − 2.05·9-s + 2.09·11-s − 0.788·13-s + 0.696·15-s + 4.76·17-s + 4.12·19-s − 2.35·21-s + 3.85·23-s − 4.48·25-s − 4.91·27-s − 6.02·29-s + 3.54·31-s + 2.03·33-s − 1.73·35-s − 0.423·37-s − 0.766·39-s + 1.23·41-s − 5.94·43-s − 1.47·45-s − 1.54·47-s − 1.12·49-s + 4.63·51-s + 6.70·53-s + 1.50·55-s + ⋯ |
L(s) = 1 | + 0.561·3-s + 0.320·5-s − 0.915·7-s − 0.684·9-s + 0.631·11-s − 0.218·13-s + 0.179·15-s + 1.15·17-s + 0.945·19-s − 0.514·21-s + 0.802·23-s − 0.897·25-s − 0.945·27-s − 1.11·29-s + 0.635·31-s + 0.354·33-s − 0.293·35-s − 0.0696·37-s − 0.122·39-s + 0.192·41-s − 0.906·43-s − 0.219·45-s − 0.225·47-s − 0.161·49-s + 0.648·51-s + 0.921·53-s + 0.202·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.251595987\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.251595987\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - 0.972T + 3T^{2} \) |
| 5 | \( 1 - 0.716T + 5T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 13 | \( 1 + 0.788T + 13T^{2} \) |
| 17 | \( 1 - 4.76T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 23 | \( 1 - 3.85T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 0.423T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 5.94T + 43T^{2} \) |
| 47 | \( 1 + 1.54T + 47T^{2} \) |
| 53 | \( 1 - 6.70T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 - 9.99T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 5.73T + 79T^{2} \) |
| 83 | \( 1 - 5.36T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 0.656T + 97T^{2} \) |
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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
−7.900208708532962735546036896579, −7.10933370565333465254123978842, −6.53370119451765180432654708842, −5.57905582639128893430644359392, −5.36182873854405502083828920141, −3.99887520594427538487733518588, −3.40383667785203826849979459355, −2.83239611507517910441483182230, −1.87503534068066487162511675295, −0.71459055495673843731905466389,
0.71459055495673843731905466389, 1.87503534068066487162511675295, 2.83239611507517910441483182230, 3.40383667785203826849979459355, 3.99887520594427538487733518588, 5.36182873854405502083828920141, 5.57905582639128893430644359392, 6.53370119451765180432654708842, 7.10933370565333465254123978842, 7.900208708532962735546036896579