| L(s) = 1 | − 4·5-s + 2·7-s − 11-s − 6·13-s + 17-s − 2·19-s + 8·23-s + 11·25-s + 2·29-s + 4·31-s − 8·35-s − 6·37-s − 2·41-s + 10·43-s − 3·49-s − 12·53-s + 4·55-s − 8·59-s + 10·61-s + 24·65-s + 16·67-s + 12·71-s − 6·73-s − 2·77-s + 10·79-s − 4·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s − 0.301·11-s − 1.66·13-s + 0.242·17-s − 0.458·19-s + 1.66·23-s + 11/5·25-s + 0.371·29-s + 0.718·31-s − 1.35·35-s − 0.986·37-s − 0.312·41-s + 1.52·43-s − 3/7·49-s − 1.64·53-s + 0.539·55-s − 1.04·59-s + 1.28·61-s + 2.97·65-s + 1.95·67-s + 1.42·71-s − 0.702·73-s − 0.227·77-s + 1.12·79-s − 0.439·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9515843294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9515843294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.24380376822227, −14.80165147511035, −14.43367440180785, −13.82580174993395, −12.87227280146540, −12.40918839917196, −12.18354418760533, −11.49163566294161, −10.92914639158973, −10.74996768923654, −9.770198525554662, −9.263980290919123, −8.422444457967529, −8.050341205147543, −7.668734190607936, −6.918447077590535, −6.714894180530025, −5.281515458453874, −5.035420159519414, −4.434206002140871, −3.846569926568190, −2.974692736807905, −2.520239826911488, −1.324221173907138, −0.4055911567167684,
0.4055911567167684, 1.324221173907138, 2.520239826911488, 2.974692736807905, 3.846569926568190, 4.434206002140871, 5.035420159519414, 5.281515458453874, 6.714894180530025, 6.918447077590535, 7.668734190607936, 8.050341205147543, 8.422444457967529, 9.263980290919123, 9.770198525554662, 10.74996768923654, 10.92914639158973, 11.49163566294161, 12.18354418760533, 12.40918839917196, 12.87227280146540, 13.82580174993395, 14.43367440180785, 14.80165147511035, 15.24380376822227
