| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 9-s − 4·14-s + 16-s + 18-s − 16·23-s + 25-s + 4·28-s − 4·29-s − 32-s − 36-s + 12·37-s + 8·43-s + 16·46-s + 9·49-s − 50-s + 12·53-s − 4·56-s + 4·58-s − 4·63-s + 64-s + 16·71-s + 72-s − 12·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1/3·9-s − 1.06·14-s + 1/4·16-s + 0.235·18-s − 3.33·23-s + 1/5·25-s + 0.755·28-s − 0.742·29-s − 0.176·32-s − 1/6·36-s + 1.97·37-s + 1.21·43-s + 2.35·46-s + 9/7·49-s − 0.141·50-s + 1.64·53-s − 0.534·56-s + 0.525·58-s − 0.503·63-s + 1/8·64-s + 1.89·71-s + 0.117·72-s − 1.39·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.458592310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458592310\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
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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
−8.226725337740102959629524523667, −7.980344033763464398860649542247, −7.59296314412578541138444157209, −7.25064516319549393530407507627, −6.45924889661775162738843244117, −6.02781434229962039022006516262, −5.72270602020186491729983018957, −5.18205850071318733136996494951, −4.53983211573212372162016618671, −4.00245154231127494487912730387, −3.71429990238753460633999447454, −2.52355701010995816854499099757, −2.25825024622411137204182017376, −1.62994298299946254523920149393, −0.67513256959857991408450373561,
0.67513256959857991408450373561, 1.62994298299946254523920149393, 2.25825024622411137204182017376, 2.52355701010995816854499099757, 3.71429990238753460633999447454, 4.00245154231127494487912730387, 4.53983211573212372162016618671, 5.18205850071318733136996494951, 5.72270602020186491729983018957, 6.02781434229962039022006516262, 6.45924889661775162738843244117, 7.25064516319549393530407507627, 7.59296314412578541138444157209, 7.980344033763464398860649542247, 8.226725337740102959629524523667
