| L(s) = 1 | − i·3-s + (0.311 + 2.21i)5-s − 4.90i·7-s − 9-s + 11-s − 4.14i·13-s + (2.21 − 0.311i)15-s − 5.33i·17-s − 5.18·19-s − 4.90·21-s + 4i·23-s + (−4.80 + 1.37i)25-s + i·27-s − 1.80·29-s − 2.62·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.139 + 0.990i)5-s − 1.85i·7-s − 0.333·9-s + 0.301·11-s − 1.15i·13-s + (0.571 − 0.0803i)15-s − 1.29i·17-s − 1.18·19-s − 1.06·21-s + 0.834i·23-s + (−0.961 + 0.275i)25-s + 0.192i·27-s − 0.335·29-s − 0.470·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8813955548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8813955548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-0.311 - 2.21i)T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 4.90iT - 7T^{2} \) |
| 13 | \( 1 + 4.14iT - 13T^{2} \) |
| 17 | \( 1 + 5.33iT - 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 5.80iT - 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 - 4.90iT - 43T^{2} \) |
| 47 | \( 1 + 7.05iT - 47T^{2} \) |
| 53 | \( 1 - 7.18iT - 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 - 0.755T + 61T^{2} \) |
| 67 | \( 1 - 4.85iT - 67T^{2} \) |
| 71 | \( 1 + 0.428T + 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 + 6.42T + 79T^{2} \) |
| 83 | \( 1 + 2.90iT - 83T^{2} \) |
| 89 | \( 1 + 0.622T + 89T^{2} \) |
| 97 | \( 1 + 2.75iT - 97T^{2} \) |
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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
−8.190603013709827337523653058503, −7.53628579320893297039126451593, −7.07104601831783766970802977765, −6.45878661838847249585147021706, −5.51618257470267948394476853436, −4.39892290537778238486795716724, −3.52303173437102395124376152919, −2.77514270505209182235749349670, −1.46287185118712345549711804443, −0.27579647668228536578355633447,
1.77532473541198652605608151932, 2.39371914054203298518144493601, 3.86091782906617814168235339385, 4.46594944128980453040777266346, 5.40661109872207471851989109025, 5.96322073996326127071579241637, 6.67463345187058765259299951920, 8.186505105761158373685278597371, 8.600327972004237773635333529809, 9.139802904578349493196110997163
