| L(s) = 1 | − 1.76e3·7-s + 1.26e4·13-s − 1.43e4·19-s − 7.81e4·25-s − 1.78e5·31-s − 6.15e5·37-s − 1.03e6·43-s + 2.28e6·49-s + 1.53e6·61-s + 4.05e6·67-s + 1.23e6·73-s + 4.24e6·79-s − 2.22e7·91-s + 5.27e6·97-s + 2.19e7·103-s − 1.68e7·109-s + ⋯ |
| L(s) = 1 | − 1.94·7-s + 1.59·13-s − 0.480·19-s − 25-s − 1.07·31-s − 1.99·37-s − 1.98·43-s + 2.77·49-s + 0.867·61-s + 1.64·67-s + 0.372·73-s + 0.968·79-s − 3.09·91-s + 0.586·97-s + 1.97·103-s − 1.24·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.096124024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096124024\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + p^{7} T^{2} \) |
| 7 | \( 1 + 1763 T + p^{7} T^{2} \) |
| 11 | \( 1 + p^{7} T^{2} \) |
| 13 | \( 1 - 12605 T + p^{7} T^{2} \) |
| 17 | \( 1 + p^{7} T^{2} \) |
| 19 | \( 1 + 14357 T + p^{7} T^{2} \) |
| 23 | \( 1 + p^{7} T^{2} \) |
| 29 | \( 1 + p^{7} T^{2} \) |
| 31 | \( 1 + 178916 T + p^{7} T^{2} \) |
| 37 | \( 1 + 615373 T + p^{7} T^{2} \) |
| 41 | \( 1 + p^{7} T^{2} \) |
| 43 | \( 1 + 1035224 T + p^{7} T^{2} \) |
| 47 | \( 1 + p^{7} T^{2} \) |
| 53 | \( 1 + p^{7} T^{2} \) |
| 59 | \( 1 + p^{7} T^{2} \) |
| 61 | \( 1 - 1537199 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4058455 T + p^{7} T^{2} \) |
| 71 | \( 1 + p^{7} T^{2} \) |
| 73 | \( 1 - 1236809 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4245427 T + p^{7} T^{2} \) |
| 83 | \( 1 + p^{7} T^{2} \) |
| 89 | \( 1 + p^{7} T^{2} \) |
| 97 | \( 1 - 5276357 T + p^{7} T^{2} \) |
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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
−9.974891998376260513473661872924, −9.127494300266366477832711833712, −8.333011118968204954267500079519, −6.94123863062516559648144196432, −6.35460587042056395637589240590, −5.46883023374237092544236254040, −3.76715408587636492182967847776, −3.37567524083087126953799579250, −1.90717305793420141450491458932, −0.45870724545555586647332224598,
0.45870724545555586647332224598, 1.90717305793420141450491458932, 3.37567524083087126953799579250, 3.76715408587636492182967847776, 5.46883023374237092544236254040, 6.35460587042056395637589240590, 6.94123863062516559648144196432, 8.333011118968204954267500079519, 9.127494300266366477832711833712, 9.974891998376260513473661872924
