| L(s) = 1 | + 2·2-s + 4-s − 2·7-s − 10·11-s − 2·13-s − 4·14-s + 16-s − 2·17-s + 4·19-s − 20·22-s − 12·23-s − 4·26-s − 2·28-s + 6·29-s − 6·31-s − 2·32-s − 4·34-s + 12·37-s + 8·38-s + 4·43-s − 10·44-s − 24·46-s + 10·47-s − 9·49-s − 2·52-s + 6·53-s + 12·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.755·7-s − 3.01·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 4.26·22-s − 2.50·23-s − 0.784·26-s − 0.377·28-s + 1.11·29-s − 1.07·31-s − 0.353·32-s − 0.685·34-s + 1.97·37-s + 1.29·38-s + 0.609·43-s − 1.50·44-s − 3.53·46-s + 1.45·47-s − 9/7·49-s − 0.277·52-s + 0.824·53-s + 1.57·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 117 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 181 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 77 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 190 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.227917741047130835531171008129, −8.093428644699479019118071719341, −7.84275254605367055434269287083, −7.41188368199710160235840706074, −7.15959164529818573222489993682, −6.47456989275849264712008695043, −6.02960198216223383698806824073, −5.79732457118863190687436351787, −5.35085230113033872625237680225, −5.15952749498294618303200234954, −4.51974133154942580013262990524, −4.50592277852821359772367514858, −3.68520609737977153625952479056, −3.62420316522075978186564613862, −2.67337915842800646791362991727, −2.62113086018999568075694296407, −2.30925725011639520550285942249, −1.29513420931283488646988174066, 0, 0,
1.29513420931283488646988174066, 2.30925725011639520550285942249, 2.62113086018999568075694296407, 2.67337915842800646791362991727, 3.62420316522075978186564613862, 3.68520609737977153625952479056, 4.50592277852821359772367514858, 4.51974133154942580013262990524, 5.15952749498294618303200234954, 5.35085230113033872625237680225, 5.79732457118863190687436351787, 6.02960198216223383698806824073, 6.47456989275849264712008695043, 7.15959164529818573222489993682, 7.41188368199710160235840706074, 7.84275254605367055434269287083, 8.093428644699479019118071719341, 8.227917741047130835531171008129
