| L(s) = 1 | − 14·3-s − 18·7-s + 98·9-s + 252·21-s − 134·23-s − 378·27-s + 612·29-s + 594·43-s − 602·47-s + 162·49-s − 1.76e3·63-s − 1.09e3·67-s + 1.87e3·69-s + 251·81-s + 154·83-s − 8.56e3·87-s − 2.77e3·89-s − 756·101-s − 2.64e3·103-s − 442·107-s − 2.66e3·121-s + 127-s − 8.31e3·129-s + 131-s + 137-s + 139-s + 8.42e3·141-s + ⋯ |
| L(s) = 1 | − 2.69·3-s − 0.971·7-s + 3.62·9-s + 2.61·21-s − 1.21·23-s − 2.69·27-s + 3.91·29-s + 2.10·43-s − 1.86·47-s + 0.472·49-s − 3.52·63-s − 2.00·67-s + 3.27·69-s + 0.344·81-s + 0.203·83-s − 10.5·87-s − 3.30·89-s − 0.744·101-s − 2.53·103-s − 0.399·107-s − 2·121-s + 0.000698·127-s − 5.67·129-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.03·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 134 T + 8978 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 306 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 74338 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 594 T + 176418 T^{2} - 594 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 602 T + 181202 T^{2} + 602 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 452342 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1098 T + 602802 T^{2} + 1098 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T + 11858 T^{2} - 154 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1386 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
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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
−9.843793468558168467740768141180, −9.598496039081116197097180137503, −8.791873130240302564114347326178, −8.398494632412803378676961684892, −7.88140100415785314858468957853, −7.25614478151514303450426943562, −6.60320901499457661957472492358, −6.59947875026069205621554412768, −6.05905876754360656004394905071, −5.92468466338773291617714966358, −5.15282062620040149458430425516, −4.99958247131198918555662384359, −4.20623904874380482717792496637, −4.11713957347578463943351156022, −2.89194778919852650489326717125, −2.69985357930334609001843806263, −1.34296313056487185585076265782, −0.979881847225025387591293291526, 0, 0,
0.979881847225025387591293291526, 1.34296313056487185585076265782, 2.69985357930334609001843806263, 2.89194778919852650489326717125, 4.11713957347578463943351156022, 4.20623904874380482717792496637, 4.99958247131198918555662384359, 5.15282062620040149458430425516, 5.92468466338773291617714966358, 6.05905876754360656004394905071, 6.59947875026069205621554412768, 6.60320901499457661957472492358, 7.25614478151514303450426943562, 7.88140100415785314858468957853, 8.398494632412803378676961684892, 8.791873130240302564114347326178, 9.598496039081116197097180137503, 9.843793468558168467740768141180
