L(s) = 1 | − 4-s + 2·9-s − 2·11-s + 16-s + 8·19-s + 12·29-s − 20·31-s − 2·36-s − 24·41-s + 2·44-s − 49-s + 12·59-s − 8·61-s − 64-s + 24·71-s − 8·76-s − 16·79-s − 5·81-s − 36·89-s − 4·99-s − 4·109-s − 12·116-s + 3·121-s + 20·124-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 2/3·9-s − 0.603·11-s + 1/4·16-s + 1.83·19-s + 2.22·29-s − 3.59·31-s − 1/3·36-s − 3.74·41-s + 0.301·44-s − 1/7·49-s + 1.56·59-s − 1.02·61-s − 1/8·64-s + 2.84·71-s − 0.917·76-s − 1.80·79-s − 5/9·81-s − 3.81·89-s − 0.402·99-s − 0.383·109-s − 1.11·116-s + 3/11·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.221197746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221197746\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.985425999812591775857981830845, −8.412190586380507843156572965288, −8.061146837409206861311940968066, −7.42876011783554666035557562147, −7.23345929626212748966836536876, −6.81558203958049030034712086005, −6.76819731821225583985873233538, −5.90568131516190423655155581512, −5.60161142452796111599114351892, −5.13133380071392062450343136350, −5.11193981738037520493063668035, −4.61575154933212041110754030755, −3.88009763141889101053290266785, −3.79382735738498953232004012135, −3.12487244622707465681327377196, −2.97833875110423585315257832570, −2.17813940416829836819236984710, −1.57092755178095351879557476020, −1.27180840645154899122025656630, −0.33711306904813403695903105835,
0.33711306904813403695903105835, 1.27180840645154899122025656630, 1.57092755178095351879557476020, 2.17813940416829836819236984710, 2.97833875110423585315257832570, 3.12487244622707465681327377196, 3.79382735738498953232004012135, 3.88009763141889101053290266785, 4.61575154933212041110754030755, 5.11193981738037520493063668035, 5.13133380071392062450343136350, 5.60161142452796111599114351892, 5.90568131516190423655155581512, 6.76819731821225583985873233538, 6.81558203958049030034712086005, 7.23345929626212748966836536876, 7.42876011783554666035557562147, 8.061146837409206861311940968066, 8.412190586380507843156572965288, 8.985425999812591775857981830845