| L(s) = 1 | + 24·5-s − 77·7-s − 408·11-s + 89·13-s + 2.08e3·17-s + 2.61e3·19-s − 1.75e3·23-s − 2.54e3·25-s − 7.29e3·29-s − 2.34e3·31-s − 1.84e3·35-s − 4.99e3·37-s − 6.52e3·41-s + 6.23e3·43-s + 2.98e4·47-s − 1.08e4·49-s + 2.26e4·53-s − 9.79e3·55-s − 1.96e4·59-s − 2.20e4·61-s + 2.13e3·65-s − 4.81e4·67-s − 5.11e4·71-s + 3.07e4·73-s + 3.14e4·77-s − 3.82e4·79-s − 8.11e3·83-s + ⋯ |
| L(s) = 1 | + 0.429·5-s − 0.593·7-s − 1.01·11-s + 0.146·13-s + 1.75·17-s + 1.66·19-s − 0.690·23-s − 0.815·25-s − 1.61·29-s − 0.438·31-s − 0.254·35-s − 0.599·37-s − 0.606·41-s + 0.513·43-s + 1.96·47-s − 0.647·49-s + 1.10·53-s − 0.436·55-s − 0.733·59-s − 0.758·61-s + 0.0627·65-s − 1.30·67-s − 1.20·71-s + 0.675·73-s + 0.603·77-s − 0.688·79-s − 0.129·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 24 T + p^{5} T^{2} \) |
| 7 | \( 1 + 11 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 408 T + p^{5} T^{2} \) |
| 13 | \( 1 - 89 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2088 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2617 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1752 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7296 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2348 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4993 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6528 T + p^{5} T^{2} \) |
| 43 | \( 1 - 6232 T + p^{5} T^{2} \) |
| 47 | \( 1 - 29832 T + p^{5} T^{2} \) |
| 53 | \( 1 - 22608 T + p^{5} T^{2} \) |
| 59 | \( 1 + 19608 T + p^{5} T^{2} \) |
| 61 | \( 1 + 22045 T + p^{5} T^{2} \) |
| 67 | \( 1 + 48131 T + p^{5} T^{2} \) |
| 71 | \( 1 + 720 p T + p^{5} T^{2} \) |
| 73 | \( 1 - 30737 T + p^{5} T^{2} \) |
| 79 | \( 1 + 38219 T + p^{5} T^{2} \) |
| 83 | \( 1 + 8112 T + p^{5} T^{2} \) |
| 89 | \( 1 + 44280 T + p^{5} T^{2} \) |
| 97 | \( 1 + 136651 T + p^{5} T^{2} \) |
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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
−9.887500466601379365849680898496, −9.221887273369369012426754036815, −7.84029942134148737383315893838, −7.31630887538678126223539827544, −5.74255480939522068170299681168, −5.47580159480496910489616341672, −3.77207300139185038407015696495, −2.84094970910369067547327850605, −1.44858435541110173930039928327, 0,
1.44858435541110173930039928327, 2.84094970910369067547327850605, 3.77207300139185038407015696495, 5.47580159480496910489616341672, 5.74255480939522068170299681168, 7.31630887538678126223539827544, 7.84029942134148737383315893838, 9.221887273369369012426754036815, 9.887500466601379365849680898496
