| L(s) = 1 | + 1.74·2-s − 3-s + 1.05·4-s + 5-s − 1.74·6-s − 3.24·7-s − 1.64·8-s + 9-s + 1.74·10-s − 4.40·11-s − 1.05·12-s + 1.48·13-s − 5.68·14-s − 15-s − 4.99·16-s − 2.48·17-s + 1.74·18-s + 0.0384·19-s + 1.05·20-s + 3.24·21-s − 7.69·22-s − 4.16·23-s + 1.64·24-s + 25-s + 2.60·26-s − 27-s − 3.43·28-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 0.577·3-s + 0.528·4-s + 0.447·5-s − 0.713·6-s − 1.22·7-s − 0.582·8-s + 0.333·9-s + 0.552·10-s − 1.32·11-s − 0.305·12-s + 0.412·13-s − 1.51·14-s − 0.258·15-s − 1.24·16-s − 0.603·17-s + 0.412·18-s + 0.00882·19-s + 0.236·20-s + 0.709·21-s − 1.64·22-s − 0.867·23-s + 0.336·24-s + 0.200·25-s + 0.510·26-s − 0.192·27-s − 0.649·28-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
| \(L(1)\) |
\(\approx\) |
\(1.780579537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780579537\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.74T + 2T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 - 0.0384T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 - 4.13T + 41T^{2} \) |
| 43 | \( 1 + 7.97T + 43T^{2} \) |
| 47 | \( 1 - 0.459T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 - 4.51T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 2.11T + 89T^{2} \) |
| 97 | \( 1 + 6.49T + 97T^{2} \) |
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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
−8.033283295674927825795764462769, −6.89268098134926088400336406450, −6.40125193984112606993582271717, −5.89100958204549069274438161407, −5.23070602704910374453287112429, −4.54253273248445439281618144818, −3.76753374832601680428160787282, −2.88941103869644865658687242281, −2.30579084791622025362421191979, −0.56382447152953755534512026771,
0.56382447152953755534512026771, 2.30579084791622025362421191979, 2.88941103869644865658687242281, 3.76753374832601680428160787282, 4.54253273248445439281618144818, 5.23070602704910374453287112429, 5.89100958204549069274438161407, 6.40125193984112606993582271717, 6.89268098134926088400336406450, 8.033283295674927825795764462769
