L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 11-s − 12-s + 3·13-s + 14-s + 16-s − 18-s + 8·19-s + 21-s + 22-s + 4·23-s + 24-s − 3·26-s − 27-s − 28-s + 2·29-s + 5·31-s − 32-s + 33-s + 36-s + 6·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.218·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 0.588·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.898·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.731625409\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.731625409\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.60329281926753, −12.18544242307703, −11.62875135388578, −11.17931569430466, −10.92722980606692, −10.41778266761964, −9.891579783348494, −9.395905228282954, −9.204296496663827, −8.575846044944063, −7.931560637601076, −7.579902687491734, −7.210972338260773, −6.564326180862957, −6.065913842460901, −5.806663293300604, −5.171544740787458, −4.616780694957645, −4.047706268634288, −3.358077978750168, −2.830825415385368, −2.458477649088600, −1.311479076680491, −1.128782812445598, −0.4808464024391496,
0.4808464024391496, 1.128782812445598, 1.311479076680491, 2.458477649088600, 2.830825415385368, 3.358077978750168, 4.047706268634288, 4.616780694957645, 5.171544740787458, 5.806663293300604, 6.065913842460901, 6.564326180862957, 7.210972338260773, 7.579902687491734, 7.931560637601076, 8.575846044944063, 9.204296496663827, 9.395905228282954, 9.891579783348494, 10.41778266761964, 10.92722980606692, 11.17931569430466, 11.62875135388578, 12.18544242307703, 12.60329281926753