| L(s) = 1 | + 3-s + 4-s − 7-s − 2·9-s + 12-s + 5·13-s + 16-s + 19-s − 21-s − 25-s − 5·27-s − 28-s + 10·31-s − 2·36-s − 2·37-s + 5·39-s + 7·43-s + 48-s − 6·49-s + 5·52-s + 57-s + 7·61-s + 2·63-s + 64-s − 2·67-s + 25·73-s − 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.377·7-s − 2/3·9-s + 0.288·12-s + 1.38·13-s + 1/4·16-s + 0.229·19-s − 0.218·21-s − 1/5·25-s − 0.962·27-s − 0.188·28-s + 1.79·31-s − 1/3·36-s − 0.328·37-s + 0.800·39-s + 1.06·43-s + 0.144·48-s − 6/7·49-s + 0.693·52-s + 0.132·57-s + 0.896·61-s + 0.251·63-s + 1/8·64-s − 0.244·67-s + 2.92·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.403133378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.403133378\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.755998672121756923337519923551, −8.325547040555248476287962151857, −7.955041684777455904918140396756, −7.66678890104364625254612085318, −6.66680069206815761329647210205, −6.62167295134143630670021221103, −6.09678251270102728990414275815, −5.48626502240429928342136496852, −5.08497044900019805772554046035, −4.10638856428906716513184363000, −3.78227202503164159946541574655, −3.08039486178299684163372214979, −2.67683103927765594767690823027, −1.90013515668153608365653766161, −0.905286876574988152666229265388,
0.905286876574988152666229265388, 1.90013515668153608365653766161, 2.67683103927765594767690823027, 3.08039486178299684163372214979, 3.78227202503164159946541574655, 4.10638856428906716513184363000, 5.08497044900019805772554046035, 5.48626502240429928342136496852, 6.09678251270102728990414275815, 6.62167295134143630670021221103, 6.66680069206815761329647210205, 7.66678890104364625254612085318, 7.955041684777455904918140396756, 8.325547040555248476287962151857, 8.755998672121756923337519923551
