| L(s) = 1 | + 2·3-s − 5-s + 3·9-s + 3·11-s − 2·15-s − 17-s + 4·27-s + 6·33-s − 3·45-s − 47-s − 2·51-s − 3·55-s − 79-s + 5·81-s − 4·83-s + 85-s + 9·99-s + 3·103-s + 109-s + 5·121-s + 125-s + 127-s + 131-s − 4·135-s + 137-s + 139-s − 2·141-s + ⋯ |
| L(s) = 1 | + 2·3-s − 5-s + 3·9-s + 3·11-s − 2·15-s − 17-s + 4·27-s + 6·33-s − 3·45-s − 47-s − 2·51-s − 3·55-s − 79-s + 5·81-s − 4·83-s + 85-s + 9·99-s + 3·103-s + 109-s + 5·121-s + 125-s + 127-s + 131-s − 4·135-s + 137-s + 139-s − 2·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.125283487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125283487\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.806825816990092411257097996474, −8.798075650899100572427306528230, −8.497468466747847048119359402158, −8.231537740057144946109243380183, −7.45585712100381803933033484258, −7.42889359140379229927584884810, −6.86870168533582490888301251589, −6.85091025212163870059722545502, −6.08652159782164190955730794349, −6.05983772566350068398653590926, −4.92551559818686070207301007065, −4.49191871455614392733831801572, −4.33946747214507812036513364147, −3.84050777907815855017148173275, −3.61591526553344854696197201096, −3.27083385335708461783193319284, −2.65251541909391063465016194480, −2.07059464360970192121691681923, −1.51837560538580053316749051678, −1.16608275555854712624366762590,
1.16608275555854712624366762590, 1.51837560538580053316749051678, 2.07059464360970192121691681923, 2.65251541909391063465016194480, 3.27083385335708461783193319284, 3.61591526553344854696197201096, 3.84050777907815855017148173275, 4.33946747214507812036513364147, 4.49191871455614392733831801572, 4.92551559818686070207301007065, 6.05983772566350068398653590926, 6.08652159782164190955730794349, 6.85091025212163870059722545502, 6.86870168533582490888301251589, 7.42889359140379229927584884810, 7.45585712100381803933033484258, 8.231537740057144946109243380183, 8.497468466747847048119359402158, 8.798075650899100572427306528230, 8.806825816990092411257097996474
