| L(s) = 1 | − 2.39·2-s + 3-s + 3.74·4-s − 5-s − 2.39·6-s + 3.91·7-s − 4.18·8-s + 9-s + 2.39·10-s + 1.97·11-s + 3.74·12-s − 1.35·13-s − 9.37·14-s − 15-s + 2.53·16-s + 4.15·17-s − 2.39·18-s + 7.85·19-s − 3.74·20-s + 3.91·21-s − 4.73·22-s + 1.26·23-s − 4.18·24-s + 25-s + 3.25·26-s + 27-s + 14.6·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.87·4-s − 0.447·5-s − 0.978·6-s + 1.47·7-s − 1.47·8-s + 0.333·9-s + 0.757·10-s + 0.595·11-s + 1.08·12-s − 0.376·13-s − 2.50·14-s − 0.258·15-s + 0.633·16-s + 1.00·17-s − 0.564·18-s + 1.80·19-s − 0.837·20-s + 0.853·21-s − 1.00·22-s + 0.263·23-s − 0.853·24-s + 0.200·25-s + 0.637·26-s + 0.192·27-s + 2.76·28-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
| \(L(1)\) |
\(\approx\) |
\(1.557937536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557937536\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.39T + 2T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 11 | \( 1 - 1.97T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 - 7.85T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 - 8.62T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 5.47T + 43T^{2} \) |
| 47 | \( 1 + 2.45T + 47T^{2} \) |
| 53 | \( 1 - 6.21T + 53T^{2} \) |
| 59 | \( 1 + 7.36T + 59T^{2} \) |
| 61 | \( 1 - 2.25T + 61T^{2} \) |
| 67 | \( 1 + 2.76T + 67T^{2} \) |
| 71 | \( 1 + 9.59T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 4.84T + 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−8.002191111065092972535002530824, −7.71178013989261588072612557322, −7.21688733167093295740603741521, −6.28844276111875075643976773938, −5.14614962928861378966536600729, −4.48234243011219710697478627778, −3.31290600515834975796688494749, −2.51247707010377939157729439477, −1.38484393426955385876857528201, −0.983385121547620145248647721192,
0.983385121547620145248647721192, 1.38484393426955385876857528201, 2.51247707010377939157729439477, 3.31290600515834975796688494749, 4.48234243011219710697478627778, 5.14614962928861378966536600729, 6.28844276111875075643976773938, 7.21688733167093295740603741521, 7.71178013989261588072612557322, 8.002191111065092972535002530824
