| L(s) = 1 | − 5.65·2-s + 6.30·3-s + 23.9·4-s − 35.6·6-s − 90.2·8-s + 12.7·9-s + 151.·12-s − 12.8·13-s + 318.·16-s − 72.2·18-s − 569.·24-s − 125·25-s + 72.5·26-s − 89.7·27-s − 257.·29-s + 196.·31-s − 1.07e3·32-s + 306.·36-s − 80.9·39-s − 478.·41-s + 532.·47-s + 2.00e3·48-s − 343·49-s + 706.·50-s − 307.·52-s + 507.·54-s + 1.45e3·58-s + ⋯ |
| L(s) = 1 | − 1.99·2-s + 1.21·3-s + 2.99·4-s − 2.42·6-s − 3.98·8-s + 0.473·9-s + 3.63·12-s − 0.273·13-s + 4.97·16-s − 0.945·18-s − 4.83·24-s − 25-s + 0.547·26-s − 0.639·27-s − 1.64·29-s + 1.14·31-s − 5.95·32-s + 1.41·36-s − 0.332·39-s − 1.82·41-s + 1.65·47-s + 6.03·48-s − 49-s + 1.99·50-s − 0.820·52-s + 1.27·54-s + 3.29·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + 5.65T + 8T^{2} \) |
| 3 | \( 1 - 6.30T + 27T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 29 | \( 1 + 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 + 478.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 - 532.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 396T + 2.05e5T^{2} \) |
| 61 | \( 1 + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 396.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 413.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−9.650167525094752719665546526069, −9.142727367362008080403396216415, −8.273854839377873230986918143317, −7.75673856470330812448552301857, −6.90344699600709196382596827894, −5.77099939202915742058881016114, −3.53834289405068283551636060737, −2.52707784573468122735832833954, −1.62477027138258993685381313017, 0,
1.62477027138258993685381313017, 2.52707784573468122735832833954, 3.53834289405068283551636060737, 5.77099939202915742058881016114, 6.90344699600709196382596827894, 7.75673856470330812448552301857, 8.273854839377873230986918143317, 9.142727367362008080403396216415, 9.650167525094752719665546526069
