L(s) = 1 | − 5.29·11-s + 5.29·23-s − 5·25-s + 10.5·29-s + 6·37-s − 12·43-s + 10.5·53-s − 4·67-s − 5.29·71-s − 8·79-s − 5.29·107-s − 18·109-s − 21.1·113-s + ⋯ |
L(s) = 1 | − 1.59·11-s + 1.10·23-s − 25-s + 1.96·29-s + 0.986·37-s − 1.82·43-s + 1.45·53-s − 0.488·67-s − 0.627·71-s − 0.900·79-s − 0.511·107-s − 1.72·109-s − 1.99·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 5.29T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.68046967421742912538433333907, −6.92368915862081844712231084358, −6.20791150142134488291686373787, −5.34127051865836217130018379303, −4.88953720900795964389067858473, −4.00703789716805152785371293092, −2.94123026085743864659582152393, −2.49393734490330670576404426053, −1.23892001217160548574782403808, 0,
1.23892001217160548574782403808, 2.49393734490330670576404426053, 2.94123026085743864659582152393, 4.00703789716805152785371293092, 4.88953720900795964389067858473, 5.34127051865836217130018379303, 6.20791150142134488291686373787, 6.92368915862081844712231084358, 7.68046967421742912538433333907