L(s) = 1 | + 2-s − 3-s + 4-s − 0.625·5-s − 6-s − 1.55·7-s + 8-s + 9-s − 0.625·10-s + 6.08·11-s − 12-s − 3.75·13-s − 1.55·14-s + 0.625·15-s + 16-s + 17-s + 18-s + 8.54·19-s − 0.625·20-s + 1.55·21-s + 6.08·22-s + 6.91·23-s − 24-s − 4.60·25-s − 3.75·26-s − 27-s − 1.55·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.279·5-s − 0.408·6-s − 0.587·7-s + 0.353·8-s + 0.333·9-s − 0.197·10-s + 1.83·11-s − 0.288·12-s − 1.04·13-s − 0.415·14-s + 0.161·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.96·19-s − 0.139·20-s + 0.339·21-s + 1.29·22-s + 1.44·23-s − 0.204·24-s − 0.921·25-s − 0.736·26-s − 0.192·27-s − 0.293·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.576571658\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.576571658\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 0.625T + 5T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 11 | \( 1 - 6.08T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 19 | \( 1 - 8.54T + 19T^{2} \) |
| 23 | \( 1 - 6.91T + 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 31 | \( 1 + 6.80T + 31T^{2} \) |
| 37 | \( 1 - 4.59T + 37T^{2} \) |
| 41 | \( 1 - 0.331T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 - 9.68T + 47T^{2} \) |
| 53 | \( 1 + 7.57T + 53T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 2.59T + 89T^{2} \) |
| 97 | \( 1 + 4.27T + 97T^{2} \) |
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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
−7.53251302300697355856053345559, −7.38283587256744218081877810961, −6.57308881379857392922605040721, −5.90427306049572045933023852060, −5.21044556253114542390973976408, −4.49115030269286657560369573330, −3.61419068615359503221026129206, −3.14637789278469144950850453153, −1.81323836647756392802692023626, −0.806869583759140423853089603070,
0.806869583759140423853089603070, 1.81323836647756392802692023626, 3.14637789278469144950850453153, 3.61419068615359503221026129206, 4.49115030269286657560369573330, 5.21044556253114542390973976408, 5.90427306049572045933023852060, 6.57308881379857392922605040721, 7.38283587256744218081877810961, 7.53251302300697355856053345559