| L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s + 9-s + 2·10-s − 4·13-s − 16-s + 2·17-s + 18-s − 2·20-s + 2·25-s − 4·26-s + 12·29-s + 5·32-s + 2·34-s − 36-s − 6·40-s − 10·41-s + 2·45-s + 11·49-s + 2·50-s + 4·52-s + 12·58-s − 6·61-s + 7·64-s − 8·65-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.10·13-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s + 2/5·25-s − 0.784·26-s + 2.22·29-s + 0.883·32-s + 0.342·34-s − 1/6·36-s − 0.948·40-s − 1.56·41-s + 0.298·45-s + 11/7·49-s + 0.282·50-s + 0.554·52-s + 1.57·58-s − 0.768·61-s + 7/8·64-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.531860069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.531860069\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.407150924620135108384091258738, −7.87318133085482548358072429913, −7.35164867954586296885765315889, −6.82585831503475015664356821287, −6.51222342351933370789282933734, −5.80133240678306495037443986095, −5.66122201885673848330428428879, −5.00565703692069895633573862956, −4.58493997257763896508957140388, −4.36258822803035292984780411632, −3.44526489943131006283279176192, −3.02143676600588145517565522278, −2.46845300944666985627378934525, −1.71177217093433097692672924897, −0.71686718238719714976558401533,
0.71686718238719714976558401533, 1.71177217093433097692672924897, 2.46845300944666985627378934525, 3.02143676600588145517565522278, 3.44526489943131006283279176192, 4.36258822803035292984780411632, 4.58493997257763896508957140388, 5.00565703692069895633573862956, 5.66122201885673848330428428879, 5.80133240678306495037443986095, 6.51222342351933370789282933734, 6.82585831503475015664356821287, 7.35164867954586296885765315889, 7.87318133085482548358072429913, 8.407150924620135108384091258738
