| L(s) = 1 | − 5-s − 5·7-s + 5·11-s + 4·13-s + 6·17-s − 2·19-s − 6·23-s + 5·25-s + 6·29-s − 5·31-s + 5·35-s + 2·37-s − 16·41-s − 8·43-s + 4·47-s + 18·49-s + 9·53-s − 5·55-s − 3·59-s + 12·61-s − 4·65-s − 2·67-s + 16·71-s + 14·73-s − 25·77-s − 79-s − 34·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s + 1.50·11-s + 1.10·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 25-s + 1.11·29-s − 0.898·31-s + 0.845·35-s + 0.328·37-s − 2.49·41-s − 1.21·43-s + 0.583·47-s + 18/7·49-s + 1.23·53-s − 0.674·55-s − 0.390·59-s + 1.53·61-s − 0.496·65-s − 0.244·67-s + 1.89·71-s + 1.63·73-s − 2.84·77-s − 0.112·79-s − 3.73·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358371760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358371760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.03588423748998573518481150375, −10.72272262441247440820082923254, −10.06504481196471554017482556095, −9.795498277928988663560234940374, −9.592779530101334843839807688943, −8.812730461102558982882346671003, −8.386693243709894395002389890294, −8.357897454813457825740921238065, −7.33901180689111044745534199649, −6.86946924839618668978276118211, −6.60937876214017638836146963732, −6.20259217815736382622553604487, −5.65879996575559199879377903551, −5.10631714560288618052034387382, −4.08445819281867512974779820046, −3.77954060820177748723425598295, −3.44287738734565821501010357804, −2.82758947239247120621671795459, −1.69489218550883750994371345228, −0.73584844925906169318956550176,
0.73584844925906169318956550176, 1.69489218550883750994371345228, 2.82758947239247120621671795459, 3.44287738734565821501010357804, 3.77954060820177748723425598295, 4.08445819281867512974779820046, 5.10631714560288618052034387382, 5.65879996575559199879377903551, 6.20259217815736382622553604487, 6.60937876214017638836146963732, 6.86946924839618668978276118211, 7.33901180689111044745534199649, 8.357897454813457825740921238065, 8.386693243709894395002389890294, 8.812730461102558982882346671003, 9.592779530101334843839807688943, 9.795498277928988663560234940374, 10.06504481196471554017482556095, 10.72272262441247440820082923254, 11.03588423748998573518481150375
