L(s) = 1 | + 3-s + 5-s + 9-s + 15-s − 2·17-s + 23-s + 25-s + 27-s + 6·29-s + 8·31-s + 10·37-s − 2·43-s + 45-s − 6·47-s − 2·51-s + 14·53-s + 14·59-s + 6·61-s − 10·67-s + 69-s + 12·71-s + 14·73-s + 75-s + 14·79-s + 81-s − 12·83-s − 2·85-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.258·15-s − 0.485·17-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.64·37-s − 0.304·43-s + 0.149·45-s − 0.875·47-s − 0.280·51-s + 1.92·53-s + 1.82·59-s + 0.768·61-s − 1.22·67-s + 0.120·69-s + 1.42·71-s + 1.63·73-s + 0.115·75-s + 1.57·79-s + 1/9·81-s − 1.31·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.679287416\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.679287416\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.58390112750769, −12.95578242476276, −12.68785908544384, −12.01373986254597, −11.44661521996636, −11.18096933993817, −10.29212012538445, −10.06139204609137, −9.687164886127687, −8.972699572954493, −8.593517317257897, −8.169067430532882, −7.633793072680604, −6.965643029938057, −6.513825429822661, −6.141184913315421, −5.346879340371283, −4.868312347106565, −4.316686490979176, −3.754206413359380, −3.070608760348842, −2.413845290935383, −2.191512537972379, −1.125291062889090, −0.7015781698706037,
0.7015781698706037, 1.125291062889090, 2.191512537972379, 2.413845290935383, 3.070608760348842, 3.754206413359380, 4.316686490979176, 4.868312347106565, 5.346879340371283, 6.141184913315421, 6.513825429822661, 6.965643029938057, 7.633793072680604, 8.169067430532882, 8.593517317257897, 8.972699572954493, 9.687164886127687, 10.06139204609137, 10.29212012538445, 11.18096933993817, 11.44661521996636, 12.01373986254597, 12.68785908544384, 12.95578242476276, 13.58390112750769