L(s) = 1 | − 0.305·3-s − 4.28·5-s − 1.13·7-s − 2.90·9-s − 5.82·11-s − 0.783·13-s + 1.30·15-s + 8.01·17-s − 0.508·19-s + 0.347·21-s − 5.66·23-s + 13.3·25-s + 1.80·27-s + 0.322·29-s + 8.64·31-s + 1.77·33-s + 4.87·35-s + 5.90·37-s + 0.239·39-s + 5.71·41-s − 5.49·43-s + 12.4·45-s − 9.12·47-s − 5.70·49-s − 2.44·51-s − 8.24·53-s + 24.9·55-s + ⋯ |
L(s) = 1 | − 0.176·3-s − 1.91·5-s − 0.429·7-s − 0.968·9-s − 1.75·11-s − 0.217·13-s + 0.337·15-s + 1.94·17-s − 0.116·19-s + 0.0758·21-s − 1.18·23-s + 2.67·25-s + 0.347·27-s + 0.0598·29-s + 1.55·31-s + 0.309·33-s + 0.823·35-s + 0.971·37-s + 0.0383·39-s + 0.891·41-s − 0.838·43-s + 1.85·45-s − 1.33·47-s − 0.815·49-s − 0.342·51-s − 1.13·53-s + 3.36·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5054370177\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5054370177\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 83 | \( 1 - T \) |
good | 3 | \( 1 + 0.305T + 3T^{2} \) |
| 5 | \( 1 + 4.28T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 + 0.783T + 13T^{2} \) |
| 17 | \( 1 - 8.01T + 17T^{2} \) |
| 19 | \( 1 + 0.508T + 19T^{2} \) |
| 23 | \( 1 + 5.66T + 23T^{2} \) |
| 29 | \( 1 - 0.322T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 + 9.12T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 - 0.322T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 + 9.07T + 71T^{2} \) |
| 73 | \( 1 + 7.73T + 73T^{2} \) |
| 79 | \( 1 + 9.84T + 79T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−9.826769182268593764345387215492, −8.354652778798753949071763404527, −8.025897732038423218612604782293, −7.53694553540144254440710277050, −6.30807885498971638065460570910, −5.34638406552120639606652814047, −4.52255592564234624126157210679, −3.34396243404857860506062632313, −2.83685743193095395074643589278, −0.49393591591599712571787153096,
0.49393591591599712571787153096, 2.83685743193095395074643589278, 3.34396243404857860506062632313, 4.52255592564234624126157210679, 5.34638406552120639606652814047, 6.30807885498971638065460570910, 7.53694553540144254440710277050, 8.025897732038423218612604782293, 8.354652778798753949071763404527, 9.826769182268593764345387215492