| L(s) = 1 | + 3·3-s − 4·7-s + 9·9-s − 28·11-s + 16·13-s − 108·17-s + 32·19-s − 12·21-s + 28·23-s + 27·27-s − 238·29-s − 180·31-s − 84·33-s + 40·37-s + 48·39-s + 422·41-s − 276·43-s − 60·47-s − 327·49-s − 324·51-s − 220·53-s + 96·57-s − 804·59-s − 358·61-s − 36·63-s + 884·67-s + 84·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.215·7-s + 1/3·9-s − 0.767·11-s + 0.341·13-s − 1.54·17-s + 0.386·19-s − 0.124·21-s + 0.253·23-s + 0.192·27-s − 1.52·29-s − 1.04·31-s − 0.443·33-s + 0.177·37-s + 0.197·39-s + 1.60·41-s − 0.978·43-s − 0.186·47-s − 0.953·49-s − 0.889·51-s − 0.570·53-s + 0.223·57-s − 1.77·59-s − 0.751·61-s − 0.0719·63-s + 1.61·67-s + 0.146·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 32 T + p^{3} T^{2} \) |
| 23 | \( 1 - 28 T + p^{3} T^{2} \) |
| 29 | \( 1 + 238 T + p^{3} T^{2} \) |
| 31 | \( 1 + 180 T + p^{3} T^{2} \) |
| 37 | \( 1 - 40 T + p^{3} T^{2} \) |
| 41 | \( 1 - 422 T + p^{3} T^{2} \) |
| 43 | \( 1 + 276 T + p^{3} T^{2} \) |
| 47 | \( 1 + 60 T + p^{3} T^{2} \) |
| 53 | \( 1 + 220 T + p^{3} T^{2} \) |
| 59 | \( 1 + 804 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 + 64 T + p^{3} T^{2} \) |
| 73 | \( 1 - 152 T + p^{3} T^{2} \) |
| 79 | \( 1 + 932 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1292 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 824 T + p^{3} T^{2} \) |
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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
−9.606041959271606658966612882925, −9.053376607574669113302260972431, −8.053320785885798280105855430855, −7.26888407315958638187784521246, −6.25153923009292657527614240521, −5.12128531316628657736918625667, −4.01684099671830750480560330534, −2.92542145344147177219298147670, −1.80395618286295856706996375198, 0,
1.80395618286295856706996375198, 2.92542145344147177219298147670, 4.01684099671830750480560330534, 5.12128531316628657736918625667, 6.25153923009292657527614240521, 7.26888407315958638187784521246, 8.053320785885798280105855430855, 9.053376607574669113302260972431, 9.606041959271606658966612882925
