| L(s) = 1 | − 4-s − 9-s − 8·11-s + 16-s + 8·19-s − 4·29-s + 8·31-s + 36-s + 16·41-s + 8·44-s + 10·49-s + 12·59-s + 28·61-s − 64-s + 4·71-s − 8·76-s + 81-s + 4·89-s + 8·99-s + 16·101-s + 28·109-s + 4·116-s + 26·121-s − 8·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 2.41·11-s + 1/4·16-s + 1.83·19-s − 0.742·29-s + 1.43·31-s + 1/6·36-s + 2.49·41-s + 1.20·44-s + 10/7·49-s + 1.56·59-s + 3.58·61-s − 1/8·64-s + 0.474·71-s − 0.917·76-s + 1/9·81-s + 0.423·89-s + 0.804·99-s + 1.59·101-s + 2.68·109-s + 0.371·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.078593783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078593783\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−8.883734070865607093545011730055, −8.819731164713835960317903770403, −8.262509490808275606086979037705, −7.892292402500814383671431939756, −7.62953912248559245935096759250, −7.37192899681605490894887814631, −6.93338551956464417723385448032, −6.29344026794985491995552079206, −5.84467766369760306801212006407, −5.42251122589078439659760444592, −5.25632789793723211254706713620, −4.98115030578720377870554750709, −4.27594077610886542899878596084, −3.92772055593496786690940044664, −3.34172748604609181180386642868, −2.84373588692901210370802614156, −2.46396108908156555758593884457, −2.10908779075671772190505074111, −0.797064277997278402066762760625, −0.70867531533723461617089106513,
0.70867531533723461617089106513, 0.797064277997278402066762760625, 2.10908779075671772190505074111, 2.46396108908156555758593884457, 2.84373588692901210370802614156, 3.34172748604609181180386642868, 3.92772055593496786690940044664, 4.27594077610886542899878596084, 4.98115030578720377870554750709, 5.25632789793723211254706713620, 5.42251122589078439659760444592, 5.84467766369760306801212006407, 6.29344026794985491995552079206, 6.93338551956464417723385448032, 7.37192899681605490894887814631, 7.62953912248559245935096759250, 7.892292402500814383671431939756, 8.262509490808275606086979037705, 8.819731164713835960317903770403, 8.883734070865607093545011730055
