| L(s) = 1 | + 4·2-s + 12·4-s − 10·5-s + 32·8-s + 8·9-s − 40·10-s + 26·13-s + 80·16-s + 32·18-s − 120·20-s + 75·25-s + 104·26-s − 64·29-s + 192·32-s + 96·36-s − 112·37-s − 320·40-s − 80·45-s + 98·49-s + 300·50-s + 312·52-s − 256·58-s + 224·61-s + 448·64-s − 260·65-s + 256·72-s + 32·73-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s + 8/9·9-s − 4·10-s + 2·13-s + 5·16-s + 16/9·18-s − 6·20-s + 3·25-s + 4·26-s − 2.20·29-s + 6·32-s + 8/3·36-s − 3.02·37-s − 8·40-s − 1.77·45-s + 2·49-s + 6·50-s + 6·52-s − 4.41·58-s + 3.67·61-s + 7·64-s − 4·65-s + 32/9·72-s + 0.438·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.006907394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.006907394\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 8 T^{2} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 488 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1048 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 1688 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 1592 T^{2} + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6728 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 112 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8872 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
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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
−12.03195378804269654193017782979, −11.73810248398557917911795360833, −11.15459127832538811505176635723, −10.91211182958322664858766981340, −10.59325953085620009386696946041, −9.924301764585639578353454773367, −8.827463154559083807253331986697, −8.433602100670123457933941526938, −7.906180538445507527383898947413, −7.17618494879587828614895114667, −7.07091945706753848924570767543, −6.60801668131805008660196530655, −5.60451817494096769850914208570, −5.39334948664616495691826605252, −4.50711214988440696130600357421, −3.93552171760998745298219858650, −3.67327604694651949628133913086, −3.37167980128748758538181347208, −2.09882385154680776884893798897, −1.11801747379473080500538273614,
1.11801747379473080500538273614, 2.09882385154680776884893798897, 3.37167980128748758538181347208, 3.67327604694651949628133913086, 3.93552171760998745298219858650, 4.50711214988440696130600357421, 5.39334948664616495691826605252, 5.60451817494096769850914208570, 6.60801668131805008660196530655, 7.07091945706753848924570767543, 7.17618494879587828614895114667, 7.906180538445507527383898947413, 8.433602100670123457933941526938, 8.827463154559083807253331986697, 9.924301764585639578353454773367, 10.59325953085620009386696946041, 10.91211182958322664858766981340, 11.15459127832538811505176635723, 11.73810248398557917911795360833, 12.03195378804269654193017782979
