| L(s) = 1 | + 3.88·2-s + 1.79·3-s + 11.0·4-s + 6.96·6-s − 7·7-s + 27.5·8-s − 5.78·9-s + 19.8·12-s + 0.743·13-s − 27.1·14-s + 62.5·16-s + 30.1·17-s − 22.4·18-s − 37.2·19-s − 12.5·21-s − 40.5·23-s + 49.3·24-s + 25·25-s + 2.88·26-s − 26.5·27-s − 77.5·28-s + 132.·32-s + 117.·34-s − 64.0·36-s − 61.3·37-s − 144.·38-s + 1.33·39-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 0.598·3-s + 2.77·4-s + 1.16·6-s − 7-s + 3.43·8-s − 0.642·9-s + 1.65·12-s + 0.0571·13-s − 1.94·14-s + 3.90·16-s + 1.77·17-s − 1.24·18-s − 1.96·19-s − 0.598·21-s − 1.76·23-s + 2.05·24-s + 25-s + 0.111·26-s − 0.982·27-s − 2.77·28-s + 4.14·32-s + 3.44·34-s − 1.77·36-s − 1.65·37-s − 3.80·38-s + 0.0341·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.432205367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.432205367\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 - 3.88T + 4T^{2} \) |
| 3 | \( 1 - 1.79T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 0.743T + 169T^{2} \) |
| 17 | \( 1 - 30.1T + 289T^{2} \) |
| 19 | \( 1 + 37.2T + 361T^{2} \) |
| 23 | \( 1 + 40.5T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 61.3T + 1.36e3T^{2} \) |
| 43 | \( 1 + 8.84T + 1.84e3T^{2} \) |
| 47 | \( 1 - 93.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 36.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 13.3T + 9.40e3T^{2} \) |
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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
−12.30302870004808925789287957158, −10.87779166020914461192141186666, −10.07740734837812492444443154499, −8.499645163769269645646988349978, −7.35002207173593457593003084242, −6.23057042167963980393176644974, −5.60077882017353762638357039311, −4.10207606168342537845641808861, −3.29425159204719963136178765038, −2.27525055906270898754169847478,
2.27525055906270898754169847478, 3.29425159204719963136178765038, 4.10207606168342537845641808861, 5.60077882017353762638357039311, 6.23057042167963980393176644974, 7.35002207173593457593003084242, 8.499645163769269645646988349978, 10.07740734837812492444443154499, 10.87779166020914461192141186666, 12.30302870004808925789287957158
