L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 13-s + 15-s + 6·17-s + 4·19-s − 4·21-s + 8·23-s + 25-s + 27-s − 4·35-s + 12·37-s + 39-s − 10·41-s − 4·43-s + 45-s + 6·47-s + 9·49-s + 6·51-s + 6·53-s + 4·57-s − 4·59-s − 6·61-s − 4·63-s + 65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.676·35-s + 1.97·37-s + 0.160·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.503·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.100824922\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.100824922\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.15121986791660, −12.70701970483265, −12.41826052107886, −11.65640815137387, −11.37804526645908, −10.53451641335073, −10.04942681431086, −9.915082485098090, −9.340243942506406, −8.909400663253500, −8.544757297333528, −7.683015583791633, −7.374882283522126, −6.941852847582767, −6.179617006236826, −6.020663776081855, −5.280623137377194, −4.840744066195990, −4.027873506410966, −3.409154886662468, −3.042742834145199, −2.782689747571903, −1.846595949141486, −1.095886593539659, −0.6292043843658601,
0.6292043843658601, 1.095886593539659, 1.846595949141486, 2.782689747571903, 3.042742834145199, 3.409154886662468, 4.027873506410966, 4.840744066195990, 5.280623137377194, 6.020663776081855, 6.179617006236826, 6.941852847582767, 7.374882283522126, 7.683015583791633, 8.544757297333528, 8.909400663253500, 9.340243942506406, 9.915082485098090, 10.04942681431086, 10.53451641335073, 11.37804526645908, 11.65640815137387, 12.41826052107886, 12.70701970483265, 13.15121986791660