| L(s) = 1 | − 4·5-s − 27·9-s + 92·13-s + 94·17-s − 109·25-s + 284·29-s + 396·37-s + 230·41-s + 108·45-s − 343·49-s + 572·53-s − 468·61-s − 368·65-s + 1.09e3·73-s + 729·81-s − 376·85-s − 1.67e3·89-s − 594·97-s − 1.94e3·101-s − 1.46e3·109-s + 2.00e3·113-s − 2.48e3·117-s + ⋯ |
| L(s) = 1 | − 0.357·5-s − 9-s + 1.96·13-s + 1.34·17-s − 0.871·25-s + 1.81·29-s + 1.75·37-s + 0.876·41-s + 0.357·45-s − 49-s + 1.48·53-s − 0.982·61-s − 0.702·65-s + 1.76·73-s + 81-s − 0.479·85-s − 1.98·89-s − 0.621·97-s − 1.91·101-s − 1.28·109-s + 1.66·113-s − 1.96·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.705981414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.705981414\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 92 T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 284 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 396 T + p^{3} T^{2} \) |
| 41 | \( 1 - 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 572 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 468 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + 1670 T + p^{3} T^{2} \) |
| 97 | \( 1 + 594 T + p^{3} T^{2} \) |
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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
−11.50745062235760894294318826877, −10.81052646260543247340368767454, −9.634851170993791615834849113203, −8.465421100951715957398218219550, −7.923524068093471190730929634931, −6.36801398443267256293315837039, −5.60618889657209613155951139928, −4.04338334463341904273363934787, −2.95402768389815526455087509694, −0.991506038462113219667934455238,
0.991506038462113219667934455238, 2.95402768389815526455087509694, 4.04338334463341904273363934787, 5.60618889657209613155951139928, 6.36801398443267256293315837039, 7.923524068093471190730929634931, 8.465421100951715957398218219550, 9.634851170993791615834849113203, 10.81052646260543247340368767454, 11.50745062235760894294318826877
