| L(s) = 1 | − 2·3-s + 3·9-s + 13-s + 4·17-s + 6·25-s − 4·27-s − 12·29-s − 2·39-s + 8·43-s − 10·49-s − 8·51-s − 20·53-s − 28·61-s − 12·75-s − 32·79-s + 5·81-s + 24·87-s + 20·101-s − 16·103-s + 24·107-s + 12·113-s + 3·117-s − 6·121-s + 127-s − 16·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 0.277·13-s + 0.970·17-s + 6/5·25-s − 0.769·27-s − 2.22·29-s − 0.320·39-s + 1.21·43-s − 1.42·49-s − 1.12·51-s − 2.74·53-s − 3.58·61-s − 1.38·75-s − 3.60·79-s + 5/9·81-s + 2.57·87-s + 1.99·101-s − 1.57·103-s + 2.32·107-s + 1.12·113-s + 0.277·117-s − 0.545·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 316368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 316368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.580724142503201408372822295574, −7.965645572263157998665195857010, −7.41436113322903083619792232153, −7.33786979345918290465240439010, −6.56187477446535398428072205989, −5.96574308578394529827741213382, −5.89249065254499898615542236081, −5.25740148441675486260967461538, −4.52069642117284203069038296829, −4.45984308910786813852795903758, −3.27214076659366118813611445645, −3.19863829716626425048202036633, −1.85452706859319256244886425142, −1.27672999999431842115430478653, 0,
1.27672999999431842115430478653, 1.85452706859319256244886425142, 3.19863829716626425048202036633, 3.27214076659366118813611445645, 4.45984308910786813852795903758, 4.52069642117284203069038296829, 5.25740148441675486260967461538, 5.89249065254499898615542236081, 5.96574308578394529827741213382, 6.56187477446535398428072205989, 7.33786979345918290465240439010, 7.41436113322903083619792232153, 7.965645572263157998665195857010, 8.580724142503201408372822295574
