L(s) = 1 | + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.707 + 0.707i)5-s − 1.84i·7-s + (0.923 − 0.382i)8-s + i·9-s + (−0.382 − 0.923i)10-s + (0.707 − 0.707i)11-s + (−0.541 − 0.541i)13-s + (1.70 + 0.707i)14-s + i·16-s + 1.84·17-s + (−0.923 − 0.382i)18-s + 20-s + (0.382 + 0.923i)22-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.707 + 0.707i)5-s − 1.84i·7-s + (0.923 − 0.382i)8-s + i·9-s + (−0.382 − 0.923i)10-s + (0.707 − 0.707i)11-s + (−0.541 − 0.541i)13-s + (1.70 + 0.707i)14-s + i·16-s + 1.84·17-s + (−0.923 − 0.382i)18-s + 20-s + (0.382 + 0.923i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7120506501\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7120506501\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 11 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + 1.84iT - T^{2} \) |
| 13 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 17 | \( 1 - 1.84T + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - 1.41T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1 + i)T - iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.765iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29752154388522028406835799432, −9.776678140510424837112753940838, −8.151651182168801929461394055838, −7.87689761170313445718726770200, −7.15577734342217511917470174196, −6.38263986196593042949494939888, −5.16024065228519847963263106007, −4.18940969103091778322272049377, −3.29061130509206823617740344142, −0.995039889397388829472766082363,
1.39683265301736417181630808059, 2.75031483247032031675647151043, 3.77537645517098113868996397896, 4.79997127995778685458258412799, 5.73103890918529311385509401574, 7.06393406803218427529422938434, 8.175342620734923029777182036544, 8.780695315834252777662822812177, 9.497754470207648709277344520023, 9.951794230634395031984996597377