L(s) = 1 | + 3-s + 3.89·5-s − 7-s + 9-s + 3.54·11-s − 6.34·13-s + 3.89·15-s + 2.74·17-s + 0.746·19-s − 21-s − 23-s + 10.1·25-s + 27-s − 10.2·29-s + 7.03·31-s + 3.54·33-s − 3.89·35-s − 7.03·37-s − 6.34·39-s + 11.0·41-s + 9.43·43-s + 3.89·45-s + 9.78·47-s + 49-s + 2.74·51-s − 2.39·53-s + 13.7·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.74·5-s − 0.377·7-s + 0.333·9-s + 1.06·11-s − 1.76·13-s + 1.00·15-s + 0.666·17-s + 0.171·19-s − 0.218·21-s − 0.208·23-s + 2.02·25-s + 0.192·27-s − 1.90·29-s + 1.26·31-s + 0.616·33-s − 0.657·35-s − 1.15·37-s − 1.01·39-s + 1.72·41-s + 1.43·43-s + 0.580·45-s + 1.42·47-s + 0.142·49-s + 0.384·51-s − 0.329·53-s + 1.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.734836828\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.734836828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 3.89T + 5T^{2} \) |
| 11 | \( 1 - 3.54T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 - 2.74T + 17T^{2} \) |
| 19 | \( 1 - 0.746T + 19T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 + 7.03T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 - 9.78T + 47T^{2} \) |
| 53 | \( 1 + 2.39T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 5.83T + 79T^{2} \) |
| 83 | \( 1 - 5.54T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.61701691928292059128225816506, −7.29602799138631328215052231155, −6.36971393201773100983475775651, −5.86061346501011050985462171864, −5.15843055996828999182670769973, −4.32392823552072945706795611372, −3.37604965244261117652366746810, −2.46295618529030660385169932401, −2.02508263472788624612758525493, −0.962137200052548421357751631647,
0.962137200052548421357751631647, 2.02508263472788624612758525493, 2.46295618529030660385169932401, 3.37604965244261117652366746810, 4.32392823552072945706795611372, 5.15843055996828999182670769973, 5.86061346501011050985462171864, 6.36971393201773100983475775651, 7.29602799138631328215052231155, 7.61701691928292059128225816506