| L(s) = 1 | − 9·2-s + 17·4-s + 343·7-s + 423·8-s + 729·9-s + 1.96e3·11-s − 3.08e3·14-s − 4.89e3·16-s − 6.56e3·18-s − 1.76e4·22-s + 2.27e4·23-s + 5.83e3·28-s − 2.12e4·29-s + 1.69e4·32-s + 1.23e4·36-s − 1.01e5·37-s + 1.26e5·43-s + 3.33e4·44-s − 2.04e5·46-s + 1.17e5·49-s − 5.03e4·53-s + 1.45e5·56-s + 1.90e5·58-s + 2.50e5·63-s + 1.60e5·64-s + 5.39e4·67-s − 2.42e5·71-s + ⋯ |
| L(s) = 1 | − 9/8·2-s + 0.265·4-s + 7-s + 0.826·8-s + 9-s + 1.47·11-s − 9/8·14-s − 1.19·16-s − 9/8·18-s − 1.65·22-s + 1.86·23-s + 0.265·28-s − 0.870·29-s + 0.518·32-s + 0.265·36-s − 1.99·37-s + 1.59·43-s + 0.391·44-s − 2.10·46-s + 49-s − 0.338·53-s + 0.826·56-s + 0.978·58-s + 63-s + 0.612·64-s + 0.179·67-s − 0.677·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.505468153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505468153\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 + 9 T + p^{6} T^{2} \) |
| 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( 1 - 1962 T + p^{6} T^{2} \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( 1 - 22734 T + p^{6} T^{2} \) |
| 29 | \( 1 + 21222 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( 1 + 101194 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 - 126614 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( 1 + 50346 T + p^{6} T^{2} \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 - 53926 T + p^{6} T^{2} \) |
| 71 | \( 1 + 242478 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( 1 - 929378 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−11.27458980478997798340008996115, −10.54036072421872383781365389031, −9.341515849022850901502772137087, −8.815265848002092986598830121936, −7.57620396673476033871019608336, −6.84022310351593936032661302339, −5.00859433059883282736261187573, −3.95719686563469584801833236872, −1.73276481640633712775694300103, −0.964393566757617540893754953678,
0.964393566757617540893754953678, 1.73276481640633712775694300103, 3.95719686563469584801833236872, 5.00859433059883282736261187573, 6.84022310351593936032661302339, 7.57620396673476033871019608336, 8.815265848002092986598830121936, 9.341515849022850901502772137087, 10.54036072421872383781365389031, 11.27458980478997798340008996115
