L(s) = 1 | + 2.23·2-s + 3-s + 3.00·4-s + 5-s + 2.23·6-s + 2.23·8-s + 9-s + 2.23·10-s − 2.47·11-s + 3.00·12-s + 4.47·13-s + 15-s − 0.999·16-s + 2·17-s + 2.23·18-s − 6.47·19-s + 3.00·20-s − 5.52·22-s + 4·23-s + 2.23·24-s + 25-s + 10.0·26-s + 27-s − 2·29-s + 2.23·30-s − 10.4·31-s − 6.70·32-s + ⋯ |
L(s) = 1 | + 1.58·2-s + 0.577·3-s + 1.50·4-s + 0.447·5-s + 0.912·6-s + 0.790·8-s + 0.333·9-s + 0.707·10-s − 0.745·11-s + 0.866·12-s + 1.24·13-s + 0.258·15-s − 0.249·16-s + 0.485·17-s + 0.527·18-s − 1.48·19-s + 0.670·20-s − 1.17·22-s + 0.834·23-s + 0.456·24-s + 0.200·25-s + 1.96·26-s + 0.192·27-s − 0.371·29-s + 0.408·30-s − 1.88·31-s − 1.18·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.424019400\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.424019400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−10.79882100591510323742516776146, −9.503460968772031648596243341298, −8.637268732626717436843359646968, −7.61793039816774615714957160372, −6.50522120888412426729021917425, −5.82006954116919153021323280215, −4.86897409772277520659255129056, −3.87338950463562701625160266593, −3.01468372532171245832878346863, −1.91305429033021291997620793361,
1.91305429033021291997620793361, 3.01468372532171245832878346863, 3.87338950463562701625160266593, 4.86897409772277520659255129056, 5.82006954116919153021323280215, 6.50522120888412426729021917425, 7.61793039816774615714957160372, 8.637268732626717436843359646968, 9.503460968772031648596243341298, 10.79882100591510323742516776146