| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 5·11-s − 12-s − 13-s + 16-s + 7·17-s + 18-s + 7·19-s − 5·22-s + 2·23-s − 24-s − 5·25-s − 26-s − 27-s − 9·29-s + 32-s + 5·33-s + 7·34-s + 36-s + 4·37-s + 7·38-s + 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.69·17-s + 0.235·18-s + 1.60·19-s − 1.06·22-s + 0.417·23-s − 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 1.67·29-s + 0.176·32-s + 0.870·33-s + 1.20·34-s + 1/6·36-s + 0.657·37-s + 1.13·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.317797603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317797603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.115371405081115629028086912492, −7.60615750221545156651025575695, −7.12878864133921339174108898199, −5.87731466493811883044268853420, −5.45857891357611499798484714443, −5.02537791498427884362166185262, −3.84008194964688739714160762493, −3.12791931944940446156639143267, −2.13423770276560204269506181932, −0.816033436839939716934410751896,
0.816033436839939716934410751896, 2.13423770276560204269506181932, 3.12791931944940446156639143267, 3.84008194964688739714160762493, 5.02537791498427884362166185262, 5.45857891357611499798484714443, 5.87731466493811883044268853420, 7.12878864133921339174108898199, 7.60615750221545156651025575695, 8.115371405081115629028086912492
