| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s − 10·11-s + 16-s − 8·17-s − 3·18-s − 19-s − 10·22-s + 7·25-s + 32-s − 8·34-s − 3·36-s − 38-s − 4·41-s − 13·43-s − 10·44-s + 11·49-s + 7·50-s + 9·59-s + 64-s + 67-s − 8·68-s − 3·72-s + 8·73-s − 76-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 3.01·11-s + 1/4·16-s − 1.94·17-s − 0.707·18-s − 0.229·19-s − 2.13·22-s + 7/5·25-s + 0.176·32-s − 1.37·34-s − 1/2·36-s − 0.162·38-s − 0.624·41-s − 1.98·43-s − 1.50·44-s + 11/7·49-s + 0.989·50-s + 1.17·59-s + 1/8·64-s + 0.122·67-s − 0.970·68-s − 0.353·72-s + 0.936·73-s − 0.114·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−9.876804091051539612976154460678, −8.877169410565898951608445881141, −8.552369633885235484828917795120, −8.183518804587056098346936373099, −7.58972698668584338176459942327, −6.87850776749608899151857958786, −6.59022774005321072172805925347, −5.67745037457113080013736267610, −5.26445438859841328879803606308, −4.94763873617583589866675821177, −4.23433316119452020253729497705, −3.21026167208467799538175600093, −2.62458680849697150871579264021, −2.23971882548717300166050683535, 0,
2.23971882548717300166050683535, 2.62458680849697150871579264021, 3.21026167208467799538175600093, 4.23433316119452020253729497705, 4.94763873617583589866675821177, 5.26445438859841328879803606308, 5.67745037457113080013736267610, 6.59022774005321072172805925347, 6.87850776749608899151857958786, 7.58972698668584338176459942327, 8.183518804587056098346936373099, 8.552369633885235484828917795120, 8.877169410565898951608445881141, 9.876804091051539612976154460678
