| L(s) = 1 | + 3·3-s − 8·7-s + 9·9-s − 4·11-s + 6·13-s + 2·17-s + 16·19-s − 24·21-s − 60·23-s + 27·27-s − 142·29-s + 176·31-s − 12·33-s + 214·37-s + 18·39-s − 278·41-s − 68·43-s + 116·47-s − 279·49-s + 6·51-s + 350·53-s + 48·57-s − 684·59-s − 394·61-s − 72·63-s + 108·67-s − 180·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.431·7-s + 1/3·9-s − 0.109·11-s + 0.128·13-s + 0.0285·17-s + 0.193·19-s − 0.249·21-s − 0.543·23-s + 0.192·27-s − 0.909·29-s + 1.01·31-s − 0.0633·33-s + 0.950·37-s + 0.0739·39-s − 1.05·41-s − 0.241·43-s + 0.360·47-s − 0.813·49-s + 0.0164·51-s + 0.907·53-s + 0.111·57-s − 1.50·59-s − 0.826·61-s − 0.143·63-s + 0.196·67-s − 0.314·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 60 T + p^{3} T^{2} \) |
| 29 | \( 1 + 142 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 278 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 116 T + p^{3} T^{2} \) |
| 53 | \( 1 - 350 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 394 T + p^{3} T^{2} \) |
| 67 | \( 1 - 108 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 398 T + p^{3} T^{2} \) |
| 79 | \( 1 + 136 T + p^{3} T^{2} \) |
| 83 | \( 1 - 436 T + p^{3} T^{2} \) |
| 89 | \( 1 + 750 T + p^{3} T^{2} \) |
| 97 | \( 1 + 82 T + p^{3} T^{2} \) |
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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
−8.182693680591538666959096617752, −7.60676236115966867735900060715, −6.69316860285617402076303082902, −5.99278568986096849768324695264, −4.99816187385205939099420184439, −4.06896616518560573632059141234, −3.26098754478655631189795926997, −2.40186957324362766207447646118, −1.31761933060326068687825865592, 0,
1.31761933060326068687825865592, 2.40186957324362766207447646118, 3.26098754478655631189795926997, 4.06896616518560573632059141234, 4.99816187385205939099420184439, 5.99278568986096849768324695264, 6.69316860285617402076303082902, 7.60676236115966867735900060715, 8.182693680591538666959096617752
