L(s) = 1 | − 2·2-s + 2·4-s − 11-s − 6·13-s − 4·16-s + 7·17-s + 5·19-s + 2·22-s − 23-s + 12·26-s + 5·29-s + 8·31-s + 8·32-s − 14·34-s + 2·37-s − 10·38-s + 12·41-s + 11·43-s − 2·44-s + 2·46-s − 8·47-s − 12·52-s − 11·53-s − 10·58-s − 5·59-s − 7·61-s − 16·62-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.301·11-s − 1.66·13-s − 16-s + 1.69·17-s + 1.14·19-s + 0.426·22-s − 0.208·23-s + 2.35·26-s + 0.928·29-s + 1.43·31-s + 1.41·32-s − 2.40·34-s + 0.328·37-s − 1.62·38-s + 1.87·41-s + 1.67·43-s − 0.301·44-s + 0.294·46-s − 1.16·47-s − 1.66·52-s − 1.51·53-s − 1.31·58-s − 0.650·59-s − 0.896·61-s − 2.03·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.02131845567736, −13.30326796813459, −12.60006884110672, −12.23863044967397, −11.83564350891200, −11.29508961639846, −10.65194349016450, −10.22328882048751, −9.772826735857230, −9.539442174886415, −9.080601828712956, −8.275408444933365, −7.839782586709641, −7.526658666144273, −7.283555594678253, −6.355807661243245, −5.960356005865926, −5.149682591692761, −4.735524992916729, −4.208899651541694, −3.079333304341218, −2.843604439260073, −2.151179385102943, −1.228642760027035, −0.8728394757417661, 0,
0.8728394757417661, 1.228642760027035, 2.151179385102943, 2.843604439260073, 3.079333304341218, 4.208899651541694, 4.735524992916729, 5.149682591692761, 5.960356005865926, 6.355807661243245, 7.283555594678253, 7.526658666144273, 7.839782586709641, 8.275408444933365, 9.080601828712956, 9.539442174886415, 9.772826735857230, 10.22328882048751, 10.65194349016450, 11.29508961639846, 11.83564350891200, 12.23863044967397, 12.60006884110672, 13.30326796813459, 14.02131845567736