| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 4·13-s + 5·16-s − 6·17-s + 4·19-s + 8·25-s + 8·26-s + 6·32-s − 12·34-s + 8·38-s − 8·43-s − 24·47-s − 4·49-s + 16·50-s + 12·52-s + 12·53-s + 12·59-s + 7·64-s − 20·67-s − 18·68-s + 12·76-s − 12·83-s − 16·86-s + 12·89-s − 48·94-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1.10·13-s + 5/4·16-s − 1.45·17-s + 0.917·19-s + 8/5·25-s + 1.56·26-s + 1.06·32-s − 2.05·34-s + 1.29·38-s − 1.21·43-s − 3.50·47-s − 4/7·49-s + 2.26·50-s + 1.66·52-s + 1.64·53-s + 1.56·59-s + 7/8·64-s − 2.44·67-s − 2.18·68-s + 1.37·76-s − 1.31·83-s − 1.72·86-s + 1.27·89-s − 4.95·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.798446809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.798446809\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.71095873299196946807929902602, −11.64547101884586091048487322303, −11.23560613038349183612688163150, −10.78740374482027599296226437607, −10.12887937150102125084130768927, −9.893877316561355144136323494636, −8.822247766061777645127620610224, −8.759060852512127316914749627063, −8.117575358009987207641949674912, −7.35276632855017526330983145889, −6.92319546903406021392152100813, −6.43108433625513742511236370740, −6.07519175867942407536279734030, −5.30985006661113254010640178615, −4.79132294398747667870265786871, −4.45372994772562693771038770054, −3.42799961740204869177711663692, −3.29944185883980678975438836056, −2.30535454201571307102710930836, −1.39145608642984413428627627051,
1.39145608642984413428627627051, 2.30535454201571307102710930836, 3.29944185883980678975438836056, 3.42799961740204869177711663692, 4.45372994772562693771038770054, 4.79132294398747667870265786871, 5.30985006661113254010640178615, 6.07519175867942407536279734030, 6.43108433625513742511236370740, 6.92319546903406021392152100813, 7.35276632855017526330983145889, 8.117575358009987207641949674912, 8.759060852512127316914749627063, 8.822247766061777645127620610224, 9.893877316561355144136323494636, 10.12887937150102125084130768927, 10.78740374482027599296226437607, 11.23560613038349183612688163150, 11.64547101884586091048487322303, 11.71095873299196946807929902602
