| L(s) = 1 | − 3·3-s − 14·5-s + 9·9-s + 4·11-s − 54·13-s + 42·15-s + 14·17-s − 92·19-s − 152·23-s + 71·25-s − 27·27-s − 106·29-s + 144·31-s − 12·33-s + 158·37-s + 162·39-s + 390·41-s − 508·43-s − 126·45-s + 528·47-s − 42·51-s + 606·53-s − 56·55-s + 276·57-s + 364·59-s − 678·61-s + 756·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.25·5-s + 1/3·9-s + 0.109·11-s − 1.15·13-s + 0.722·15-s + 0.199·17-s − 1.11·19-s − 1.37·23-s + 0.567·25-s − 0.192·27-s − 0.678·29-s + 0.834·31-s − 0.0633·33-s + 0.702·37-s + 0.665·39-s + 1.48·41-s − 1.80·43-s − 0.417·45-s + 1.63·47-s − 0.115·51-s + 1.57·53-s − 0.137·55-s + 0.641·57-s + 0.803·59-s − 1.42·61-s + 1.44·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6990570068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6990570068\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 - 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 508 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 - 606 T + p^{3} T^{2} \) |
| 59 | \( 1 - 364 T + p^{3} T^{2} \) |
| 61 | \( 1 + 678 T + p^{3} T^{2} \) |
| 67 | \( 1 - 844 T + p^{3} T^{2} \) |
| 71 | \( 1 + 8 T + p^{3} T^{2} \) |
| 73 | \( 1 - 422 T + p^{3} T^{2} \) |
| 79 | \( 1 - 384 T + p^{3} T^{2} \) |
| 83 | \( 1 - 548 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1502 T + p^{3} T^{2} \) |
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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
−10.39697519221273219773165377423, −9.540681941358917864728672066726, −8.307119842683330085741499981555, −7.64623699531748283936543360696, −6.77040263465354876446366978891, −5.69685198427605566396711113184, −4.51524023592095186587561893035, −3.86094642201379620048109741390, −2.31271776923834928150750858762, −0.49455119663587675923333046082,
0.49455119663587675923333046082, 2.31271776923834928150750858762, 3.86094642201379620048109741390, 4.51524023592095186587561893035, 5.69685198427605566396711113184, 6.77040263465354876446366978891, 7.64623699531748283936543360696, 8.307119842683330085741499981555, 9.540681941358917864728672066726, 10.39697519221273219773165377423
