| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 3·9-s − 2·11-s − 2·13-s + 4·15-s + 6·17-s − 4·21-s + 3·25-s + 4·27-s + 16·29-s − 4·31-s − 4·33-s − 4·35-s + 8·37-s − 4·39-s + 16·41-s + 14·43-s + 6·45-s − 4·47-s + 2·49-s + 12·51-s + 4·53-s − 4·55-s − 16·59-s − 6·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 0.603·11-s − 0.554·13-s + 1.03·15-s + 1.45·17-s − 0.872·21-s + 3/5·25-s + 0.769·27-s + 2.97·29-s − 0.718·31-s − 0.696·33-s − 0.676·35-s + 1.31·37-s − 0.640·39-s + 2.49·41-s + 2.13·43-s + 0.894·45-s − 0.583·47-s + 2/7·49-s + 1.68·51-s + 0.549·53-s − 0.539·55-s − 2.08·59-s − 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6969600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6969600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.566818190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.566818190\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 202 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.302151279616999268790674874451, −8.732462468163192598020273793109, −8.181778850433677918944686002009, −7.951905579840709794025666240534, −7.58820864190194669731088043629, −7.35976193273058390720146973677, −6.70602705487532374829318980739, −6.40997546153718178503646187094, −5.96492485714835452313233742623, −5.68969889722908516659920983729, −5.14235227456467984133219723135, −4.60774043885907971676049514492, −4.35528793109392519791079201569, −3.75188361628566405920594420619, −3.08972534284097917405077690425, −2.93979649941588685509397875640, −2.42523559964561513854801814983, −2.18183591034351161673892935637, −1.11905910247211498743823245817, −0.841482612885529386570118168704,
0.841482612885529386570118168704, 1.11905910247211498743823245817, 2.18183591034351161673892935637, 2.42523559964561513854801814983, 2.93979649941588685509397875640, 3.08972534284097917405077690425, 3.75188361628566405920594420619, 4.35528793109392519791079201569, 4.60774043885907971676049514492, 5.14235227456467984133219723135, 5.68969889722908516659920983729, 5.96492485714835452313233742623, 6.40997546153718178503646187094, 6.70602705487532374829318980739, 7.35976193273058390720146973677, 7.58820864190194669731088043629, 7.951905579840709794025666240534, 8.181778850433677918944686002009, 8.732462468163192598020273793109, 9.302151279616999268790674874451
