| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s − 2·11-s + 12-s − 4·13-s + 4·14-s + 16-s − 6·17-s + 18-s + 4·19-s + 4·21-s − 2·22-s + 24-s − 4·26-s + 27-s + 4·28-s + 8·29-s + 8·31-s + 32-s − 2·33-s − 6·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.872·21-s − 0.426·22-s + 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.755·28-s + 1.48·29-s + 1.43·31-s + 0.176·32-s − 0.348·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.225792896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.225792896\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.02786041371068, −13.81017817118465, −13.05782830875833, −12.50163350396818, −12.09186649565130, −11.61637705905608, −11.01651473065207, −10.66739049721433, −10.06191693960319, −9.498441045225097, −8.832224031768519, −8.287396967027276, −7.904401030117757, −7.397243346493502, −6.859888239999237, −6.308562145563401, −5.397832530427210, −5.031228885498295, −4.509037870213113, −4.247916065881970, −3.216820527468605, −2.636371970109543, −2.230388506100974, −1.549234336926227, −0.6806727814179624,
0.6806727814179624, 1.549234336926227, 2.230388506100974, 2.636371970109543, 3.216820527468605, 4.247916065881970, 4.509037870213113, 5.031228885498295, 5.397832530427210, 6.308562145563401, 6.859888239999237, 7.397243346493502, 7.904401030117757, 8.287396967027276, 8.832224031768519, 9.498441045225097, 10.06191693960319, 10.66739049721433, 11.01651473065207, 11.61637705905608, 12.09186649565130, 12.50163350396818, 13.05782830875833, 13.81017817118465, 14.02786041371068
