L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·9-s + 4·11-s − 16-s − 2·18-s − 4·22-s + 4·23-s + 25-s − 5·32-s − 2·36-s − 20·37-s − 4·43-s − 4·44-s − 4·46-s − 50-s + 8·53-s + 7·64-s − 8·67-s − 12·71-s + 6·72-s + 20·74-s + 4·79-s − 5·81-s + 4·86-s + 12·88-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s + 1.20·11-s − 1/4·16-s − 0.471·18-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.883·32-s − 1/3·36-s − 3.28·37-s − 0.609·43-s − 0.603·44-s − 0.589·46-s − 0.141·50-s + 1.09·53-s + 7/8·64-s − 0.977·67-s − 1.42·71-s + 0.707·72-s + 2.32·74-s + 0.450·79-s − 5/9·81-s + 0.431·86-s + 1.27·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 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
−8.038959738483472522625344908113, −7.30253895220535151648636726974, −7.20732777244403505922853222891, −6.82341677731459228692565952406, −6.27335381081189810983185767619, −5.65661774889156874502643200793, −5.09173380414656901481636344705, −4.77591609071825090281162346613, −4.20648989306295640420561851054, −3.66451821544217414798686071226, −3.36198027275384227529391531819, −2.37733682211854170704863740349, −1.49432146018316215927687130478, −1.26575477722673737406895852980, 0,
1.26575477722673737406895852980, 1.49432146018316215927687130478, 2.37733682211854170704863740349, 3.36198027275384227529391531819, 3.66451821544217414798686071226, 4.20648989306295640420561851054, 4.77591609071825090281162346613, 5.09173380414656901481636344705, 5.65661774889156874502643200793, 6.27335381081189810983185767619, 6.82341677731459228692565952406, 7.20732777244403505922853222891, 7.30253895220535151648636726974, 8.038959738483472522625344908113