| L(s) = 1 | + 5·9-s − 10·11-s + 12·19-s + 18·29-s + 16·41-s − 49-s − 16·59-s − 8·61-s + 16·71-s + 6·79-s + 16·81-s + 32·89-s − 50·99-s − 28·101-s + 22·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 60·171-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 3.01·11-s + 2.75·19-s + 3.34·29-s + 2.49·41-s − 1/7·49-s − 2.08·59-s − 1.02·61-s + 1.89·71-s + 0.675·79-s + 16/9·81-s + 3.39·89-s − 5.02·99-s − 2.78·101-s + 2.10·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 4.58·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.570220864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.570220864\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.830355631542343965210369418935, −9.511146756694063072849780364616, −9.074472203004943738961237653960, −8.454319508032720097249076904898, −7.77483863312673220349810341921, −7.74152208327946008940158465062, −7.64451692575950282623666912201, −7.09863865729534740582752256860, −6.38508060345397383102242753684, −6.21194920886477618236243061206, −5.33803774843133144785432928121, −5.19725513004886246870487322093, −4.73191917989119424612534923405, −4.50306578537665513108174397465, −3.66694085477657493600548533871, −3.01517848820079930900166197188, −2.77379127881215527383193628069, −2.24087465779208734826785086378, −1.21516411902723416399131658557, −0.75426739068348235660808495487,
0.75426739068348235660808495487, 1.21516411902723416399131658557, 2.24087465779208734826785086378, 2.77379127881215527383193628069, 3.01517848820079930900166197188, 3.66694085477657493600548533871, 4.50306578537665513108174397465, 4.73191917989119424612534923405, 5.19725513004886246870487322093, 5.33803774843133144785432928121, 6.21194920886477618236243061206, 6.38508060345397383102242753684, 7.09863865729534740582752256860, 7.64451692575950282623666912201, 7.74152208327946008940158465062, 7.77483863312673220349810341921, 8.454319508032720097249076904898, 9.074472203004943738961237653960, 9.511146756694063072849780364616, 9.830355631542343965210369418935
