| L(s) = 1 | + 7.95e9·7-s − 2.32e12·13-s − 1.01e15·19-s − 1.19e16·25-s + 2.34e17·31-s − 4.33e17·37-s − 7.06e18·43-s + 3.58e19·49-s − 2.81e20·61-s − 3.80e20·67-s + 5.20e21·73-s − 1.16e22·79-s − 1.84e22·91-s + 8.71e22·97-s − 1.95e23·103-s − 4.33e23·109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.359·13-s − 1.99·19-s − 25-s + 1.65·31-s − 0.400·37-s − 1.16·43-s + 1.30·49-s − 0.826·61-s − 0.380·67-s + 1.94·73-s − 1.74·79-s − 0.545·91-s + 1.23·97-s − 1.38·103-s − 1.60·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{23} T^{2} \) |
| 7 | \( 1 - 7950445508 T + p^{23} T^{2} \) |
| 11 | \( 1 + p^{23} T^{2} \) |
| 13 | \( 1 + 2321530104406 T + p^{23} T^{2} \) |
| 17 | \( 1 + p^{23} T^{2} \) |
| 19 | \( 1 + 1015192461697768 T + p^{23} T^{2} \) |
| 23 | \( 1 + p^{23} T^{2} \) |
| 29 | \( 1 + p^{23} T^{2} \) |
| 31 | \( 1 - 234040242184219556 T + p^{23} T^{2} \) |
| 37 | \( 1 + 433556633400399010 T + p^{23} T^{2} \) |
| 41 | \( 1 + p^{23} T^{2} \) |
| 43 | \( 1 + 7068861656394413224 T + p^{23} T^{2} \) |
| 47 | \( 1 + p^{23} T^{2} \) |
| 53 | \( 1 + p^{23} T^{2} \) |
| 59 | \( 1 + p^{23} T^{2} \) |
| 61 | \( 1 + \)\(28\!\cdots\!26\)\( T + p^{23} T^{2} \) |
| 67 | \( 1 + \)\(38\!\cdots\!48\)\( T + p^{23} T^{2} \) |
| 71 | \( 1 + p^{23} T^{2} \) |
| 73 | \( 1 - \)\(52\!\cdots\!10\)\( T + p^{23} T^{2} \) |
| 79 | \( 1 + \)\(11\!\cdots\!16\)\( T + p^{23} T^{2} \) |
| 83 | \( 1 + p^{23} T^{2} \) |
| 89 | \( 1 + p^{23} T^{2} \) |
| 97 | \( 1 - \)\(87\!\cdots\!22\)\( T + p^{23} T^{2} \) |
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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
−11.22375784243986276720105291326, −10.18465421356491778288536732465, −8.622007884466173987598596508585, −7.84673387089773264913800801554, −6.41271170991065313263829698950, −5.01701022817541056125330742775, −4.13221044571701129757826738911, −2.39077415345825424200279389549, −1.46007367448076262243371775103, 0,
1.46007367448076262243371775103, 2.39077415345825424200279389549, 4.13221044571701129757826738911, 5.01701022817541056125330742775, 6.41271170991065313263829698950, 7.84673387089773264913800801554, 8.622007884466173987598596508585, 10.18465421356491778288536732465, 11.22375784243986276720105291326
