L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 4·11-s − 12-s − 13-s + 16-s + 5·17-s − 18-s − 6·19-s + 4·22-s − 5·23-s + 24-s + 26-s − 27-s − 3·29-s − 7·31-s − 32-s + 4·33-s − 5·34-s + 36-s + 4·37-s + 6·38-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 − 1.20·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 1.37·19-s + 0.852·22-s − 1.04·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.557·29-s − 1.25·31-s − 0.176·32-s + 0.696·33-s − 0.857·34-s + 1/6·36-s + 0.657·37-s + 0.973·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5639417360\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5639417360\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.80026490304047987407832038523, −7.45532249609355029973068795695, −6.51848525844095474718013006566, −5.82696431594499896967504679599, −5.31693893898766106398470087941, −4.38784012067125971630260569341, −3.49117067483915395749917424675, −2.46994279221533923406612875867, −1.71867244622787668090936495163, −0.42168321265853577669066162718,
0.42168321265853577669066162718, 1.71867244622787668090936495163, 2.46994279221533923406612875867, 3.49117067483915395749917424675, 4.38784012067125971630260569341, 5.31693893898766106398470087941, 5.82696431594499896967504679599, 6.51848525844095474718013006566, 7.45532249609355029973068795695, 7.80026490304047987407832038523