L(s) = 1 | − 3-s − 7-s − 2·9-s − 11-s − 3·17-s − 7·19-s + 21-s − 23-s + 5·27-s + 3·29-s − 8·31-s + 33-s − 37-s − 11·41-s − 11·43-s + 12·47-s − 6·49-s + 3·51-s + 6·53-s + 7·57-s + 9·59-s − 9·61-s + 2·63-s − 3·67-s + 69-s + 5·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.727·17-s − 1.60·19-s + 0.218·21-s − 0.208·23-s + 0.962·27-s + 0.557·29-s − 1.43·31-s + 0.174·33-s − 0.164·37-s − 1.71·41-s − 1.67·43-s + 1.75·47-s − 6/7·49-s + 0.420·51-s + 0.824·53-s + 0.927·57-s + 1.17·59-s − 1.15·61-s + 0.251·63-s − 0.366·67-s + 0.120·69-s + 0.593·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.45378609507649, −15.02696166705783, −14.56295857125084, −13.88393931088490, −13.28158689639970, −12.98121672003859, −12.22245764078562, −11.90849677269518, −11.22483079369126, −10.73294259895372, −10.36511175699062, −9.745422413432702, −8.847160512003694, −8.662466814864811, −8.060851429552000, −7.185014754275518, −6.607265695468352, −6.305971010221649, −5.465767965193703, −5.136537763741789, −4.290597563318360, −3.730306041733043, −2.877546796953702, −2.272107895715691, −1.444794021750973, 0, 0,
1.444794021750973, 2.272107895715691, 2.877546796953702, 3.730306041733043, 4.290597563318360, 5.136537763741789, 5.465767965193703, 6.305971010221649, 6.607265695468352, 7.185014754275518, 8.060851429552000, 8.662466814864811, 8.847160512003694, 9.745422413432702, 10.36511175699062, 10.73294259895372, 11.22483079369126, 11.90849677269518, 12.22245764078562, 12.98121672003859, 13.28158689639970, 13.88393931088490, 14.56295857125084, 15.02696166705783, 15.45378609507649