L(s) = 1 | + 1.80·3-s + 5-s + 4.65·7-s + 0.265·9-s + 4.38·11-s + 4.79·13-s + 1.80·15-s + 0.285·17-s + 3.30·19-s + 8.40·21-s − 23-s + 25-s − 4.94·27-s + 3.47·29-s − 2.44·31-s + 7.91·33-s + 4.65·35-s − 11.0·37-s + 8.66·39-s + 11.7·41-s − 2.24·43-s + 0.265·45-s − 8.28·47-s + 14.6·49-s + 0.515·51-s − 3.81·53-s + 4.38·55-s + ⋯ |
L(s) = 1 | + 1.04·3-s + 0.447·5-s + 1.75·7-s + 0.0885·9-s + 1.32·11-s + 1.32·13-s + 0.466·15-s + 0.0691·17-s + 0.759·19-s + 1.83·21-s − 0.208·23-s + 0.200·25-s − 0.950·27-s + 0.644·29-s − 0.439·31-s + 1.37·33-s + 0.786·35-s − 1.81·37-s + 1.38·39-s + 1.83·41-s − 0.343·43-s + 0.0395·45-s − 1.20·47-s + 2.09·49-s + 0.0721·51-s − 0.524·53-s + 0.590·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.964243458\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.964243458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 - 0.285T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 29 | \( 1 - 3.47T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 + 8.28T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 - 5.54T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 8.15T + 83T^{2} \) |
| 89 | \( 1 - 1.98T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−8.066250396002627896743677832823, −7.41431651848770364248179944190, −6.55398314145448694896630655519, −5.75751513496079383713513703156, −5.08488356639519425763404217113, −4.14186803774509417457090762311, −3.61140383426831443260231224735, −2.65760796394772530480496964985, −1.61093390561504521606933920247, −1.31657234338124411785084788110,
1.31657234338124411785084788110, 1.61093390561504521606933920247, 2.65760796394772530480496964985, 3.61140383426831443260231224735, 4.14186803774509417457090762311, 5.08488356639519425763404217113, 5.75751513496079383713513703156, 6.55398314145448694896630655519, 7.41431651848770364248179944190, 8.066250396002627896743677832823