| L(s) = 1 | − 3-s − 1.61·5-s + 7-s + 9-s + 3·11-s + 0.618·13-s + 1.61·15-s + 6.70·17-s + 4.23·19-s − 21-s + 23-s − 2.38·25-s − 27-s − 1.76·29-s + 3·31-s − 3·33-s − 1.61·35-s + 9.47·37-s − 0.618·39-s − 11.9·41-s + 9.09·43-s − 1.61·45-s + 9.23·47-s + 49-s − 6.70·51-s − 9.32·53-s − 4.85·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.723·5-s + 0.377·7-s + 0.333·9-s + 0.904·11-s + 0.171·13-s + 0.417·15-s + 1.62·17-s + 0.971·19-s − 0.218·21-s + 0.208·23-s − 0.476·25-s − 0.192·27-s − 0.327·29-s + 0.538·31-s − 0.522·33-s − 0.273·35-s + 1.55·37-s − 0.0989·39-s − 1.86·41-s + 1.38·43-s − 0.241·45-s + 1.34·47-s + 0.142·49-s − 0.939·51-s − 1.28·53-s − 0.654·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.824513865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.824513865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.61T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 0.618T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 9.47T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 9.32T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 - 5.85T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 - 7.47T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.81757599108416002108208553075, −7.24668686245608572676297219072, −6.48671595727903152365567502304, −5.67497380672793220589644645072, −5.18604481537598688761659513029, −4.18090917364503751735920203824, −3.73743748736139043819899988461, −2.80872153513425669464973692142, −1.46034976645961276655715718156, −0.77591689989313673412833269959,
0.77591689989313673412833269959, 1.46034976645961276655715718156, 2.80872153513425669464973692142, 3.73743748736139043819899988461, 4.18090917364503751735920203824, 5.18604481537598688761659513029, 5.67497380672793220589644645072, 6.48671595727903152365567502304, 7.24668686245608572676297219072, 7.81757599108416002108208553075
