| L(s) = 1 | − 1.61·2-s + 2.88·3-s + 0.616·4-s − 4.65·6-s − 7-s + 2.23·8-s + 5.29·9-s + 3.66·11-s + 1.77·12-s + 2.37·13-s + 1.61·14-s − 4.85·16-s + 3.07·17-s − 8.57·18-s − 1.07·19-s − 2.88·21-s − 5.92·22-s − 23-s + 6.44·24-s − 3.83·26-s + 6.62·27-s − 0.616·28-s + 4.59·29-s − 2.50·31-s + 3.37·32-s + 10.5·33-s − 4.96·34-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 1.66·3-s + 0.308·4-s − 1.90·6-s − 0.377·7-s + 0.791·8-s + 1.76·9-s + 1.10·11-s + 0.512·12-s + 0.658·13-s + 0.432·14-s − 1.21·16-s + 0.744·17-s − 2.02·18-s − 0.246·19-s − 0.628·21-s − 1.26·22-s − 0.208·23-s + 1.31·24-s − 0.753·26-s + 1.27·27-s − 0.116·28-s + 0.853·29-s − 0.449·31-s + 0.596·32-s + 1.83·33-s − 0.851·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.096208893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.096208893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 2.88T + 3T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 - 2.37T + 13T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 29 | \( 1 - 4.59T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 6.54T + 53T^{2} \) |
| 59 | \( 1 + 0.409T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 7.28T + 71T^{2} \) |
| 73 | \( 1 + 4.31T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 + 6.50T + 83T^{2} \) |
| 89 | \( 1 + 4.98T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.323556736363284406086062713390, −8.277338336019245467166619015614, −7.15097366601434162543546849551, −6.79186941196047938678581162924, −5.55673819804593643045854677383, −4.18453063503472260795153780876, −3.79922279730939896205584955958, −2.80594399012448828150931275624, −1.81677205993332852584878619513, −0.984151629059009808090479916639,
0.984151629059009808090479916639, 1.81677205993332852584878619513, 2.80594399012448828150931275624, 3.79922279730939896205584955958, 4.18453063503472260795153780876, 5.55673819804593643045854677383, 6.79186941196047938678581162924, 7.15097366601434162543546849551, 8.277338336019245467166619015614, 8.323556736363284406086062713390
