L(s) = 1 | − 2-s + 2.80·3-s + 4-s − 5-s − 2.80·6-s + 3.67·7-s − 8-s + 4.85·9-s + 10-s + 3.39·11-s + 2.80·12-s − 13-s − 3.67·14-s − 2.80·15-s + 16-s − 0.772·17-s − 4.85·18-s + 4.44·19-s − 20-s + 10.2·21-s − 3.39·22-s − 1.47·23-s − 2.80·24-s + 25-s + 26-s + 5.20·27-s + 3.67·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.61·3-s + 0.5·4-s − 0.447·5-s − 1.14·6-s + 1.38·7-s − 0.353·8-s + 1.61·9-s + 0.316·10-s + 1.02·11-s + 0.809·12-s − 0.277·13-s − 0.981·14-s − 0.723·15-s + 0.250·16-s − 0.187·17-s − 1.14·18-s + 1.02·19-s − 0.223·20-s + 2.24·21-s − 0.723·22-s − 0.306·23-s − 0.572·24-s + 0.200·25-s + 0.196·26-s + 1.00·27-s + 0.693·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.102437273\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.102437273\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 - 2.80T + 3T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 17 | \( 1 + 0.772T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 - 9.18T + 29T^{2} \) |
| 37 | \( 1 + 6.06T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 4.54T + 61T^{2} \) |
| 67 | \( 1 + 2.60T + 67T^{2} \) |
| 71 | \( 1 - 7.48T + 71T^{2} \) |
| 73 | \( 1 + 6.55T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 + 1.75T + 89T^{2} \) |
| 97 | \( 1 + 8.03T + 97T^{2} \) |
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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
−8.498030018463212698166040032553, −8.045057432631391795597851742037, −7.19732002871832191434488421408, −6.77283745496239095520285618979, −5.33221052154077377367360634496, −4.45243880809548578841893400155, −3.66981759910685037369027706511, −2.83274353549169631227676386043, −1.88970746712734143783271201697, −1.15866848263455505116978459783,
1.15866848263455505116978459783, 1.88970746712734143783271201697, 2.83274353549169631227676386043, 3.66981759910685037369027706511, 4.45243880809548578841893400155, 5.33221052154077377367360634496, 6.77283745496239095520285618979, 7.19732002871832191434488421408, 8.045057432631391795597851742037, 8.498030018463212698166040032553