| L(s) = 1 | + 3-s − 5-s − 2.68·7-s + 9-s − 3.24·11-s − 4.97·13-s − 15-s + 0.921·17-s − 1.24·19-s − 2.68·21-s − 6.57·23-s + 25-s + 27-s − 3.89·29-s − 31-s − 3.24·33-s + 2.68·35-s + 2.97·37-s − 4.97·39-s + 4·41-s − 5.65·43-s − 45-s + 8.86·47-s + 0.207·49-s + 0.921·51-s + 0.792·53-s + 3.24·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.01·7-s + 0.333·9-s − 0.977·11-s − 1.37·13-s − 0.258·15-s + 0.223·17-s − 0.284·19-s − 0.585·21-s − 1.37·23-s + 0.200·25-s + 0.192·27-s − 0.722·29-s − 0.179·31-s − 0.564·33-s + 0.453·35-s + 0.488·37-s − 0.795·39-s + 0.624·41-s − 0.862·43-s − 0.149·45-s + 1.29·47-s + 0.0295·49-s + 0.129·51-s + 0.108·53-s + 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9991057924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9991057924\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 + 4.97T + 13T^{2} \) |
| 17 | \( 1 - 0.921T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 37 | \( 1 - 2.97T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - 0.792T + 53T^{2} \) |
| 59 | \( 1 - 7.00T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 3.13T + 89T^{2} \) |
| 97 | \( 1 + 4.57T + 97T^{2} \) |
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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
−7.84454999432954529397948328200, −7.33670786425379314276733601305, −6.66417191486382046839394296494, −5.77224282334651592314317523263, −5.07388780680986382623513801921, −4.15787504389829007082193780161, −3.53405750608004708519535069635, −2.64283551797617855945049853761, −2.13165338453468838603365355271, −0.45280914384978678153324353018,
0.45280914384978678153324353018, 2.13165338453468838603365355271, 2.64283551797617855945049853761, 3.53405750608004708519535069635, 4.15787504389829007082193780161, 5.07388780680986382623513801921, 5.77224282334651592314317523263, 6.66417191486382046839394296494, 7.33670786425379314276733601305, 7.84454999432954529397948328200
