| L(s) = 1 | − 7·3-s + 5·5-s − 20·7-s + 22·9-s − 6·11-s + 47·13-s − 35·15-s − 132·17-s − 146·19-s + 140·21-s − 23·23-s + 25·25-s + 35·27-s − 99·29-s + 253·31-s + 42·33-s − 100·35-s − 118·37-s − 329·39-s + 495·41-s − 272·43-s + 110·45-s − 639·47-s + 57·49-s + 924·51-s − 342·53-s − 30·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s − 1.07·7-s + 0.814·9-s − 0.164·11-s + 1.00·13-s − 0.602·15-s − 1.88·17-s − 1.76·19-s + 1.45·21-s − 0.208·23-s + 1/5·25-s + 0.249·27-s − 0.633·29-s + 1.46·31-s + 0.221·33-s − 0.482·35-s − 0.524·37-s − 1.35·39-s + 1.88·41-s − 0.964·43-s + 0.364·45-s − 1.98·47-s + 0.166·49-s + 2.53·51-s − 0.886·53-s − 0.0735·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3672204413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3672204413\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 132 T + p^{3} T^{2} \) |
| 19 | \( 1 + 146 T + p^{3} T^{2} \) |
| 29 | \( 1 + 99 T + p^{3} T^{2} \) |
| 31 | \( 1 - 253 T + p^{3} T^{2} \) |
| 37 | \( 1 + 118 T + p^{3} T^{2} \) |
| 41 | \( 1 - 495 T + p^{3} T^{2} \) |
| 43 | \( 1 + 272 T + p^{3} T^{2} \) |
| 47 | \( 1 + 639 T + p^{3} T^{2} \) |
| 53 | \( 1 + 342 T + p^{3} T^{2} \) |
| 59 | \( 1 + 240 T + p^{3} T^{2} \) |
| 61 | \( 1 + 370 T + p^{3} T^{2} \) |
| 67 | \( 1 + 698 T + p^{3} T^{2} \) |
| 71 | \( 1 - 357 T + p^{3} T^{2} \) |
| 73 | \( 1 + 259 T + p^{3} T^{2} \) |
| 79 | \( 1 + 542 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1248 T + p^{3} T^{2} \) |
| 89 | \( 1 + 828 T + p^{3} T^{2} \) |
| 97 | \( 1 - 992 T + p^{3} T^{2} \) |
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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
−8.971954303714654727796409081580, −8.218701828120330343601384296961, −6.78891324636555814081245245085, −6.30784434373476545635965359030, −6.06989070778517505532010449235, −4.83647242148016342702801659522, −4.16159761800242376164513798675, −2.88654027613438609121176250011, −1.71940221600814986856869169955, −0.29584959184463077167035035001,
0.29584959184463077167035035001, 1.71940221600814986856869169955, 2.88654027613438609121176250011, 4.16159761800242376164513798675, 4.83647242148016342702801659522, 6.06989070778517505532010449235, 6.30784434373476545635965359030, 6.78891324636555814081245245085, 8.218701828120330343601384296961, 8.971954303714654727796409081580
