L(s) = 1 | + 5·9-s − 2·11-s − 14·31-s − 16·41-s + 10·49-s + 10·59-s + 24·61-s + 6·71-s − 20·79-s + 16·81-s − 30·89-s − 10·99-s + 4·101-s − 20·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 5/3·9-s − 0.603·11-s − 2.51·31-s − 2.49·41-s + 10/7·49-s + 1.30·59-s + 3.07·61-s + 0.712·71-s − 2.25·79-s + 16/9·81-s − 3.17·89-s − 1.00·99-s + 0.398·101-s − 1.91·109-s + 3/11·121-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 + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.013615322\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013615322\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 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
−8.665698093272711287546339429481, −8.259006030621535316934884903817, −7.81322293939355045824314336652, −7.14782282991405075375376931907, −7.08844497532134828477364697417, −7.00602941825289942251902585716, −6.59101159067835030906321698276, −5.72563659315937116859868993227, −5.51886163905214676100774541635, −5.43151271222498705066696534751, −4.75390618813271717153623928527, −4.40237965223868227791761932714, −4.01409720753106071396164710116, −3.54728046189461839761545605800, −3.38291483460000137498878470416, −2.51655034540659664564190265948, −2.18608893983310120829240641878, −1.65904108881844393081677704161, −1.23830926299751393356431473401, −0.40576330652328546375250683797,
0.40576330652328546375250683797, 1.23830926299751393356431473401, 1.65904108881844393081677704161, 2.18608893983310120829240641878, 2.51655034540659664564190265948, 3.38291483460000137498878470416, 3.54728046189461839761545605800, 4.01409720753106071396164710116, 4.40237965223868227791761932714, 4.75390618813271717153623928527, 5.43151271222498705066696534751, 5.51886163905214676100774541635, 5.72563659315937116859868993227, 6.59101159067835030906321698276, 7.00602941825289942251902585716, 7.08844497532134828477364697417, 7.14782282991405075375376931907, 7.81322293939355045824314336652, 8.259006030621535316934884903817, 8.665698093272711287546339429481