| L(s) = 1 | + 4-s − 5-s − 5·9-s + 2·11-s + 16-s − 14·19-s − 20-s + 25-s − 6·29-s − 14·31-s − 5·36-s + 12·41-s + 2·44-s + 5·45-s + 11·49-s − 2·55-s − 12·59-s − 2·61-s + 64-s + 6·71-s − 14·76-s − 20·79-s − 80-s + 16·81-s + 18·89-s + 14·95-s − 10·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.447·5-s − 5/3·9-s + 0.603·11-s + 1/4·16-s − 3.21·19-s − 0.223·20-s + 1/5·25-s − 1.11·29-s − 2.51·31-s − 5/6·36-s + 1.87·41-s + 0.301·44-s + 0.745·45-s + 11/7·49-s − 0.269·55-s − 1.56·59-s − 0.256·61-s + 1/8·64-s + 0.712·71-s − 1.60·76-s − 2.25·79-s − 0.111·80-s + 16/9·81-s + 1.90·89-s + 1.43·95-s − 1.00·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−9.515625676781913720459205161908, −8.996361573480265218453634288731, −8.766840852787028026444155939270, −8.285662434899332908926558240119, −7.56981196469027501153188760806, −7.20457953717496815600620253355, −6.43810722624715010093850437821, −5.90185767008403089888905192216, −5.73876439085829035805370640147, −4.68830133406592764908928362464, −4.01173135747517138761270457828, −3.51902803001209100282256024354, −2.52121766859740692142668949704, −1.97170842587956421114470880781, 0,
1.97170842587956421114470880781, 2.52121766859740692142668949704, 3.51902803001209100282256024354, 4.01173135747517138761270457828, 4.68830133406592764908928362464, 5.73876439085829035805370640147, 5.90185767008403089888905192216, 6.43810722624715010093850437821, 7.20457953717496815600620253355, 7.56981196469027501153188760806, 8.285662434899332908926558240119, 8.766840852787028026444155939270, 8.996361573480265218453634288731, 9.515625676781913720459205161908
