| L(s) = 1 | + 22·5-s + 18·13-s + 94·17-s + 359·25-s − 130·29-s − 214·37-s + 230·41-s − 343·49-s + 518·53-s − 830·61-s + 396·65-s + 1.09e3·73-s + 2.06e3·85-s + 1.67e3·89-s + 594·97-s + 598·101-s + 1.74e3·109-s − 2.00e3·113-s + ⋯ |
| L(s) = 1 | + 1.96·5-s + 0.384·13-s + 1.34·17-s + 2.87·25-s − 0.832·29-s − 0.950·37-s + 0.876·41-s − 49-s + 1.34·53-s − 1.74·61-s + 0.755·65-s + 1.76·73-s + 2.63·85-s + 1.98·89-s + 0.621·97-s + 0.589·101-s + 1.53·109-s − 1.66·113-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
| \(L(2)\) |
\(\approx\) |
\(3.123488414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.123488414\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 22 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 518 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 830 T + p^{3} T^{2} \) |
| 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
−10.21385742603914999117947264013, −9.516541066234962366279697932684, −8.825019425432978578278725600933, −7.60022997411387479209818687959, −6.45876988353249691840789648106, −5.75414673615616438121012304524, −5.02166477292761797219467028122, −3.39649182421221517211745551547, −2.18683682347686173971765050638, −1.17737293056312354482909877744,
1.17737293056312354482909877744, 2.18683682347686173971765050638, 3.39649182421221517211745551547, 5.02166477292761797219467028122, 5.75414673615616438121012304524, 6.45876988353249691840789648106, 7.60022997411387479209818687959, 8.825019425432978578278725600933, 9.516541066234962366279697932684, 10.21385742603914999117947264013
