| L(s) = 1 | + 6·5-s + 16·7-s − 12·11-s − 38·13-s + 126·17-s + 20·19-s + 168·23-s − 89·25-s + 30·29-s + 88·31-s + 96·35-s − 254·37-s − 42·41-s − 52·43-s − 96·47-s − 87·49-s + 198·53-s − 72·55-s + 660·59-s + 538·61-s − 228·65-s + 884·67-s + 792·71-s + 218·73-s − 192·77-s + 520·79-s + 492·83-s + ⋯ |
| L(s) = 1 | + 0.536·5-s + 0.863·7-s − 0.328·11-s − 0.810·13-s + 1.79·17-s + 0.241·19-s + 1.52·23-s − 0.711·25-s + 0.192·29-s + 0.509·31-s + 0.463·35-s − 1.12·37-s − 0.159·41-s − 0.184·43-s − 0.297·47-s − 0.253·49-s + 0.513·53-s − 0.176·55-s + 1.45·59-s + 1.12·61-s − 0.435·65-s + 1.61·67-s + 1.32·71-s + 0.349·73-s − 0.284·77-s + 0.740·79-s + 0.650·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.497060907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.497060907\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 + 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 538 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 - 792 T + p^{3} T^{2} \) |
| 73 | \( 1 - 218 T + p^{3} T^{2} \) |
| 79 | \( 1 - 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1154 T + p^{3} T^{2} \) |
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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
−10.16323327281757289459714580121, −9.647749068711201366504139561259, −8.458293005545676138869194155423, −7.70467446460717101407584223692, −6.79669373143806905955336083137, −5.40745262195262579975997084113, −5.04126545786114290007724359637, −3.49923152707152173613446899008, −2.26125926996525853202434334938, −1.00620379860836101528179305729,
1.00620379860836101528179305729, 2.26125926996525853202434334938, 3.49923152707152173613446899008, 5.04126545786114290007724359637, 5.40745262195262579975997084113, 6.79669373143806905955336083137, 7.70467446460717101407584223692, 8.458293005545676138869194155423, 9.647749068711201366504139561259, 10.16323327281757289459714580121
