| L(s) = 1 | + 3·3-s + 12·5-s + 9·9-s − 20·11-s − 84·13-s + 36·15-s − 96·17-s − 12·19-s + 176·23-s + 19·25-s + 27·27-s + 58·29-s + 264·31-s − 60·33-s + 258·37-s − 252·39-s − 156·43-s + 108·45-s + 408·47-s − 288·51-s − 722·53-s − 240·55-s − 36·57-s − 492·59-s − 492·61-s − 1.00e3·65-s − 412·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.07·5-s + 1/3·9-s − 0.548·11-s − 1.79·13-s + 0.619·15-s − 1.36·17-s − 0.144·19-s + 1.59·23-s + 0.151·25-s + 0.192·27-s + 0.371·29-s + 1.52·31-s − 0.316·33-s + 1.14·37-s − 1.03·39-s − 0.553·43-s + 0.357·45-s + 1.26·47-s − 0.790·51-s − 1.87·53-s − 0.588·55-s − 0.0836·57-s − 1.08·59-s − 1.03·61-s − 1.92·65-s − 0.751·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 84 T + p^{3} T^{2} \) |
| 17 | \( 1 + 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 12 T + p^{3} T^{2} \) |
| 23 | \( 1 - 176 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 264 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 156 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 722 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 492 T + p^{3} T^{2} \) |
| 67 | \( 1 + 412 T + p^{3} T^{2} \) |
| 71 | \( 1 + 296 T + p^{3} T^{2} \) |
| 73 | \( 1 - 240 T + p^{3} T^{2} \) |
| 79 | \( 1 + 776 T + p^{3} T^{2} \) |
| 83 | \( 1 + 924 T + p^{3} T^{2} \) |
| 89 | \( 1 + 744 T + p^{3} T^{2} \) |
| 97 | \( 1 + 168 T + p^{3} T^{2} \) |
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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
−8.304376220884243568252328354450, −7.45547194363902050219697625718, −6.76525801708657016618944527566, −5.95510354573309399101771908635, −4.87031840008146602990607312562, −4.50086211202905603931467811650, −2.74585536873939059644271309102, −2.61532557524562323753925802224, −1.45308387910270756100266535207, 0,
1.45308387910270756100266535207, 2.61532557524562323753925802224, 2.74585536873939059644271309102, 4.50086211202905603931467811650, 4.87031840008146602990607312562, 5.95510354573309399101771908635, 6.76525801708657016618944527566, 7.45547194363902050219697625718, 8.304376220884243568252328354450
