L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 4·5-s − 4·6-s − 4·8-s + 2·9-s − 8·10-s + 4·11-s + 6·12-s − 6·13-s + 8·15-s + 5·16-s + 8·17-s − 4·18-s + 6·19-s + 12·20-s − 8·22-s − 2·23-s − 8·24-s + 11·25-s + 12·26-s + 6·27-s − 16·30-s + 8·31-s − 6·32-s + 8·33-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.78·5-s − 1.63·6-s − 1.41·8-s + 2/3·9-s − 2.52·10-s + 1.20·11-s + 1.73·12-s − 1.66·13-s + 2.06·15-s + 5/4·16-s + 1.94·17-s − 0.942·18-s + 1.37·19-s + 2.68·20-s − 1.70·22-s − 0.417·23-s − 1.63·24-s + 11/5·25-s + 2.35·26-s + 1.15·27-s − 2.92·30-s + 1.43·31-s − 1.06·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.896107029\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.896107029\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.909019313289954528911472144695, −9.863799273715597252472957019026, −9.438155601454903147557820918142, −9.258725652792588385750377375437, −8.785464947559925019730791366086, −8.290260678100921602776673099097, −7.896124883047979078406510087824, −7.44142111514652985071167618170, −7.08275830103697089805721086852, −6.65620813428663314754936913889, −5.92669686253139785541748268046, −5.91966122199761279484072653741, −4.91436355291107972404741626140, −4.84733197437917578494953704639, −3.57838092787783101088530829958, −3.21436558443299442175368177187, −2.53101598790025368461818334848, −2.34352637770315376767693448059, −1.27832549824080320124244507217, −1.21245014815265443091380975368,
1.21245014815265443091380975368, 1.27832549824080320124244507217, 2.34352637770315376767693448059, 2.53101598790025368461818334848, 3.21436558443299442175368177187, 3.57838092787783101088530829958, 4.84733197437917578494953704639, 4.91436355291107972404741626140, 5.91966122199761279484072653741, 5.92669686253139785541748268046, 6.65620813428663314754936913889, 7.08275830103697089805721086852, 7.44142111514652985071167618170, 7.896124883047979078406510087824, 8.290260678100921602776673099097, 8.785464947559925019730791366086, 9.258725652792588385750377375437, 9.438155601454903147557820918142, 9.863799273715597252472957019026, 9.909019313289954528911472144695