| L(s) = 1 | − 3-s + 7-s + 9-s − 7·13-s − 4·17-s − 3·19-s − 21-s + 6·23-s − 27-s − 8·29-s + 3·31-s + 11·37-s + 7·39-s − 6·41-s + 43-s − 4·47-s + 49-s + 4·51-s + 6·53-s + 3·57-s − 6·59-s − 13·61-s + 63-s − 5·67-s − 6·69-s + 6·71-s − 3·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.94·13-s − 0.970·17-s − 0.688·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s − 1.48·29-s + 0.538·31-s + 1.80·37-s + 1.12·39-s − 0.937·41-s + 0.152·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 0.397·57-s − 0.781·59-s − 1.66·61-s + 0.125·63-s − 0.610·67-s − 0.722·69-s + 0.712·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.89370756710742, −12.74515785260994, −11.94535986648971, −11.83987869637411, −11.07710344232401, −10.95010203542826, −10.37357675022866, −9.780596972505621, −9.398099004648912, −9.033098122395318, −8.338974572598264, −7.823986670098878, −7.366846443233941, −6.921337409439224, −6.535997829317220, −5.846973434606222, −5.381327692423652, −4.852138204034713, −4.425713202788850, −4.141575973853205, −3.086707895984175, −2.703620122286921, −2.034389628404570, −1.549282847430327, −0.5975014582953245, 0,
0.5975014582953245, 1.549282847430327, 2.034389628404570, 2.703620122286921, 3.086707895984175, 4.141575973853205, 4.425713202788850, 4.852138204034713, 5.381327692423652, 5.846973434606222, 6.535997829317220, 6.921337409439224, 7.366846443233941, 7.823986670098878, 8.338974572598264, 9.033098122395318, 9.398099004648912, 9.780596972505621, 10.37357675022866, 10.95010203542826, 11.07710344232401, 11.83987869637411, 11.94535986648971, 12.74515785260994, 12.89370756710742
