| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s + 16-s − 3·18-s − 6·25-s + 32-s − 3·36-s − 12·41-s − 14·49-s − 6·50-s + 64-s − 3·72-s + 4·73-s + 9·81-s − 12·82-s − 12·89-s − 14·98-s − 6·100-s − 12·113-s − 6·121-s + 127-s + 128-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s + 1/4·16-s − 0.707·18-s − 6/5·25-s + 0.176·32-s − 1/2·36-s − 1.87·41-s − 2·49-s − 0.848·50-s + 1/8·64-s − 0.353·72-s + 0.468·73-s + 81-s − 1.32·82-s − 1.27·89-s − 1.41·98-s − 3/5·100-s − 1.12·113-s − 0.545·121-s + 0.0887·127-s + 0.0883·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.399886346774878559137174178585, −7.88684048978126381217542492577, −7.56060559614021326068327337427, −6.80651390351812872698780021460, −6.51381584271712600788625580678, −6.02912286371580766090732607860, −5.48254554868235229133511167766, −5.13666955833740913576864914592, −4.59579960004521300840001644363, −3.90545652203749007176530528818, −3.42317766598572338287219915148, −2.91058877858681728047200596914, −2.20042906850218140078251630960, −1.49842124412666087384393251972, 0,
1.49842124412666087384393251972, 2.20042906850218140078251630960, 2.91058877858681728047200596914, 3.42317766598572338287219915148, 3.90545652203749007176530528818, 4.59579960004521300840001644363, 5.13666955833740913576864914592, 5.48254554868235229133511167766, 6.02912286371580766090732607860, 6.51381584271712600788625580678, 6.80651390351812872698780021460, 7.56060559614021326068327337427, 7.88684048978126381217542492577, 8.399886346774878559137174178585
