| L(s) = 1 | − 2-s − 1.23·3-s + 4-s − 5-s + 1.23·6-s − 0.381·7-s − 8-s − 1.47·9-s + 10-s − 1.23·12-s − 0.618·13-s + 0.381·14-s + 1.23·15-s + 16-s + 0.763·17-s + 1.47·18-s − 7.09·19-s − 20-s + 0.472·21-s + 6.85·23-s + 1.23·24-s + 25-s + 0.618·26-s + 5.52·27-s − 0.381·28-s − 3.23·29-s − 1.23·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.713·3-s + 0.5·4-s − 0.447·5-s + 0.504·6-s − 0.144·7-s − 0.353·8-s − 0.490·9-s + 0.316·10-s − 0.356·12-s − 0.171·13-s + 0.102·14-s + 0.319·15-s + 0.250·16-s + 0.185·17-s + 0.346·18-s − 1.62·19-s − 0.223·20-s + 0.103·21-s + 1.42·23-s + 0.252·24-s + 0.200·25-s + 0.121·26-s + 1.06·27-s − 0.0721·28-s − 0.600·29-s − 0.225·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5795128846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5795128846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 7 | \( 1 + 0.381T + 7T^{2} \) |
| 13 | \( 1 + 0.618T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 - 5.85T + 37T^{2} \) |
| 41 | \( 1 + 4.85T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 - 8.94T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−9.753934297297095681413146683450, −8.823326044063476464685607206393, −8.270601874588487330773693815833, −7.21951551124691759693812085119, −6.53283270907535514896398735374, −5.66022680726739255361377533399, −4.70675088327309224806476965137, −3.47431637875756302287885451120, −2.28183430013606027378851758661, −0.63009409150402513985240828487,
0.63009409150402513985240828487, 2.28183430013606027378851758661, 3.47431637875756302287885451120, 4.70675088327309224806476965137, 5.66022680726739255361377533399, 6.53283270907535514896398735374, 7.21951551124691759693812085119, 8.270601874588487330773693815833, 8.823326044063476464685607206393, 9.753934297297095681413146683450
