L(s) = 1 | + (−0.707 + 0.707i)3-s + 1.41i·7-s − 1.00i·9-s + (−1 + i)13-s + (−1.00 − 1.00i)21-s + i·25-s + (0.707 + 0.707i)27-s − 1.41·31-s + (1 + i)37-s − 1.41i·39-s − 1.00·49-s + (1 − i)61-s + 1.41·63-s + (1.41 − 1.41i)67-s + (−0.707 − 0.707i)75-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + 1.41i·7-s − 1.00i·9-s + (−1 + i)13-s + (−1.00 − 1.00i)21-s + i·25-s + (0.707 + 0.707i)27-s − 1.41·31-s + (1 + i)37-s − 1.41i·39-s − 1.00·49-s + (1 − i)61-s + 1.41·63-s + (1.41 − 1.41i)67-s + (−0.707 − 0.707i)75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6551410767\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6551410767\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−10.97906695086689203669690864401, −9.626310323968048748989983933636, −9.441288557782877988434647510542, −8.466911060945603222420061089713, −7.17887634069304071626328509295, −6.24251051548040817548455498901, −5.37356384890399184672899423801, −4.70122051681237414997937803449, −3.43294835580117660338348996374, −2.10110209145205183716226362850,
0.75137044797525364260919094343, 2.38076276436362608287248420143, 3.90513823797067780262373371644, 4.95346695150744839011667156222, 5.86771939584615238294315056329, 6.97786142857710253982331009585, 7.47448400418578472126877030440, 8.235966008383804841800828033247, 9.674456822114586874000020282693, 10.45654648858001826076431688013