| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s − 12-s − 2·13-s + 4·14-s + 16-s − 2·17-s + 18-s + 19-s − 4·21-s + 8·23-s − 24-s − 2·26-s − 27-s + 4·28-s − 6·29-s + 4·31-s + 32-s − 2·34-s + 36-s + 10·37-s + 38-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 − 0.288·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.872·21-s + 1.66·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 1/6·36-s + 1.64·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.548544122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.548544122\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−12.55843348903323, −12.14303687984618, −11.52099016359220, −11.21213197462256, −11.01921193792349, −10.60973936163706, −9.861765082405542, −9.393761669655263, −8.986669775970807, −8.253471295394235, −7.875648262675451, −7.397894098854581, −7.035710222497019, −6.456876282199762, −5.887737088043556, −5.366438008152002, −5.088420014954107, −4.495759277331318, −4.272742130407878, −3.631014431917854, −2.685543070993264, −2.515681369530516, −1.714274444612363, −1.156060943639050, −0.6025494476162458,
0.6025494476162458, 1.156060943639050, 1.714274444612363, 2.515681369530516, 2.685543070993264, 3.631014431917854, 4.272742130407878, 4.495759277331318, 5.088420014954107, 5.366438008152002, 5.887737088043556, 6.456876282199762, 7.035710222497019, 7.397894098854581, 7.875648262675451, 8.253471295394235, 8.986669775970807, 9.393761669655263, 9.861765082405542, 10.60973936163706, 11.01921193792349, 11.21213197462256, 11.52099016359220, 12.14303687984618, 12.55843348903323
