L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 11-s − 12-s + 6·13-s − 15-s + 16-s + 2·17-s + 18-s − 4·19-s + 20-s + 22-s − 24-s + 25-s + 6·26-s − 27-s − 10·29-s − 30-s + 32-s − 33-s + 2·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.213·22-s − 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 1.85·29-s − 0.182·30-s + 0.176·32-s − 0.174·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.877960898\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.877960898\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.40228492181983409018356243315, −11.03006073797186653339668028089, −9.918282683247511119839364045067, −8.806615184716062235501383813449, −7.57573281855054290442967607898, −6.26970322751556902553424782587, −5.88065102738235433506263724139, −4.54831431199383754044494926069, −3.43907663083633253170978783027, −1.62793361398186118811552584678,
1.62793361398186118811552584678, 3.43907663083633253170978783027, 4.54831431199383754044494926069, 5.88065102738235433506263724139, 6.26970322751556902553424782587, 7.57573281855054290442967607898, 8.806615184716062235501383813449, 9.918282683247511119839364045067, 11.03006073797186653339668028089, 11.40228492181983409018356243315