L(s) = 1 | − 3-s − 2·7-s + 9-s − 5·11-s − 13-s + 17-s + 19-s + 2·21-s − 4·23-s − 27-s − 4·29-s − 9·31-s + 5·33-s + 9·37-s + 39-s + 43-s − 6·47-s − 3·49-s − 51-s + 2·53-s − 57-s − 2·59-s − 9·61-s − 2·63-s − 12·67-s + 4·69-s + 6·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.242·17-s + 0.229·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 0.742·29-s − 1.61·31-s + 0.870·33-s + 1.47·37-s + 0.160·39-s + 0.152·43-s − 0.875·47-s − 3/7·49-s − 0.140·51-s + 0.274·53-s − 0.132·57-s − 0.260·59-s − 1.15·61-s − 0.251·63-s − 1.46·67-s + 0.481·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.95287859663926, −12.66395500861702, −12.12615383873778, −11.69533532921994, −11.10809507846586, −10.70100029967277, −10.34812336161020, −9.823995704324091, −9.369848184552442, −9.089964035451386, −8.101069328682163, −7.902296273101113, −7.417225301600932, −6.939381943836159, −6.305540946364902, −5.806163778514132, −5.528042467045804, −4.966917102494041, −4.390520383790407, −3.852067608464883, −3.136936926689895, −2.809170934861656, −2.039906193854720, −1.513147432957477, −0.4843857273189600, 0,
0.4843857273189600, 1.513147432957477, 2.039906193854720, 2.809170934861656, 3.136936926689895, 3.852067608464883, 4.390520383790407, 4.966917102494041, 5.528042467045804, 5.806163778514132, 6.305540946364902, 6.939381943836159, 7.417225301600932, 7.902296273101113, 8.101069328682163, 9.089964035451386, 9.369848184552442, 9.823995704324091, 10.34812336161020, 10.70100029967277, 11.10809507846586, 11.69533532921994, 12.12615383873778, 12.66395500861702, 12.95287859663926