L(s) = 1 | − 0.0247·2-s − 3-s − 1.99·4-s + 5-s + 0.0247·6-s − 4.38·7-s + 0.0990·8-s + 9-s − 0.0247·10-s − 2.88·11-s + 1.99·12-s − 3.62·13-s + 0.108·14-s − 15-s + 3.99·16-s − 4.67·17-s − 0.0247·18-s + 2.40·19-s − 1.99·20-s + 4.38·21-s + 0.0713·22-s + 8.04·23-s − 0.0990·24-s + 25-s + 0.0896·26-s − 27-s + 8.76·28-s + ⋯ |
L(s) = 1 | − 0.0175·2-s − 0.577·3-s − 0.999·4-s + 0.447·5-s + 0.0101·6-s − 1.65·7-s + 0.0350·8-s + 0.333·9-s − 0.00782·10-s − 0.869·11-s + 0.577·12-s − 1.00·13-s + 0.0290·14-s − 0.258·15-s + 0.999·16-s − 1.13·17-s − 0.00583·18-s + 0.551·19-s − 0.447·20-s + 0.956·21-s + 0.0152·22-s + 1.67·23-s − 0.0202·24-s + 0.200·25-s + 0.0175·26-s − 0.192·27-s + 1.65·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 0.0247T + 2T^{2} \) |
| 7 | \( 1 + 4.38T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 - 2.40T + 19T^{2} \) |
| 23 | \( 1 - 8.04T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 + 0.379T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 4.05T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 + 3.04T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 2.67T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 - 7.64T + 97T^{2} \) |
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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
−7.55040472923253221938990795093, −6.96652131796783663593266319204, −6.24616164506259850755663403750, −5.54468927346207076844558977391, −4.86307927472566039400697146183, −4.27042072386522904699875166486, −3.07081266054550455982297718834, −2.61020387206893668640890996437, −0.900572946514806112862131392943, 0,
0.900572946514806112862131392943, 2.61020387206893668640890996437, 3.07081266054550455982297718834, 4.27042072386522904699875166486, 4.86307927472566039400697146183, 5.54468927346207076844558977391, 6.24616164506259850755663403750, 6.96652131796783663593266319204, 7.55040472923253221938990795093