| L(s) = 1 | − 2·3-s + 3·5-s + 7-s + 9-s − 3·11-s + 4·13-s − 6·15-s − 3·17-s − 2·21-s + 4·25-s + 4·27-s − 6·29-s − 4·31-s + 6·33-s + 3·35-s − 2·37-s − 8·39-s + 6·41-s + 43-s + 3·45-s + 3·47-s − 6·49-s + 6·51-s − 12·53-s − 9·55-s − 6·59-s − 61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 1.54·15-s − 0.727·17-s − 0.436·21-s + 4/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 1.04·33-s + 0.507·35-s − 0.328·37-s − 1.28·39-s + 0.937·41-s + 0.152·43-s + 0.447·45-s + 0.437·47-s − 6/7·49-s + 0.840·51-s − 1.64·53-s − 1.21·55-s − 0.781·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.69103170795787164928090099492, −6.80343192905411718058101572940, −6.05494870480703676827987196213, −5.74303400337078993307294328689, −5.11368255356697024189490674537, −4.34957971927849818682973003524, −3.14803217938274589534683531641, −2.12428535443104525506599977808, −1.35204174985454524472787523361, 0,
1.35204174985454524472787523361, 2.12428535443104525506599977808, 3.14803217938274589534683531641, 4.34957971927849818682973003524, 5.11368255356697024189490674537, 5.74303400337078993307294328689, 6.05494870480703676827987196213, 6.80343192905411718058101572940, 7.69103170795787164928090099492
