| L(s) = 1 | − 1.45·2-s − 3-s + 0.127·4-s + 5-s + 1.45·6-s + 3.69·7-s + 2.73·8-s + 9-s − 1.45·10-s + 4.35·11-s − 0.127·12-s + 13-s − 5.38·14-s − 15-s − 4.23·16-s − 7.97·17-s − 1.45·18-s − 4.24·19-s + 0.127·20-s − 3.69·21-s − 6.35·22-s + 8.86·23-s − 2.73·24-s + 25-s − 1.45·26-s − 27-s + 0.469·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.577·3-s + 0.0635·4-s + 0.447·5-s + 0.595·6-s + 1.39·7-s + 0.965·8-s + 0.333·9-s − 0.461·10-s + 1.31·11-s − 0.0367·12-s + 0.277·13-s − 1.43·14-s − 0.258·15-s − 1.05·16-s − 1.93·17-s − 0.343·18-s − 0.973·19-s + 0.0284·20-s − 0.805·21-s − 1.35·22-s + 1.84·23-s − 0.557·24-s + 0.200·25-s − 0.286·26-s − 0.192·27-s + 0.0887·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192908278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192908278\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 17 | \( 1 + 7.97T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 8.86T + 23T^{2} \) |
| 29 | \( 1 + 5.36T + 29T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 + 4.99T + 53T^{2} \) |
| 59 | \( 1 - 6.63T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 0.464T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 8.62T + 79T^{2} \) |
| 83 | \( 1 - 3.91T + 83T^{2} \) |
| 89 | \( 1 + 3.24T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.286019583068096034917412500083, −7.40838744872186118059561087214, −6.75283219336604347327839905009, −6.16869108981829241890359859740, −5.01891060354342474894061308185, −4.59770521003990648603573740821, −3.92018970573797353698156268513, −2.23288978272917779544090520468, −1.58946498546053391875332651572, −0.75220307444631755495556009702,
0.75220307444631755495556009702, 1.58946498546053391875332651572, 2.23288978272917779544090520468, 3.92018970573797353698156268513, 4.59770521003990648603573740821, 5.01891060354342474894061308185, 6.16869108981829241890359859740, 6.75283219336604347327839905009, 7.40838744872186118059561087214, 8.286019583068096034917412500083
