| L(s) = 1 | − 2·2-s + 2·4-s − 3·7-s + 3·11-s + 6·13-s + 6·14-s − 4·16-s + 3·17-s − 19-s − 6·22-s + 4·23-s − 12·26-s − 6·28-s + 10·29-s + 2·31-s + 8·32-s − 6·34-s − 8·37-s + 2·38-s + 8·41-s + 43-s + 6·44-s − 8·46-s + 3·47-s + 2·49-s + 12·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.13·7-s + 0.904·11-s + 1.66·13-s + 1.60·14-s − 16-s + 0.727·17-s − 0.229·19-s − 1.27·22-s + 0.834·23-s − 2.35·26-s − 1.13·28-s + 1.85·29-s + 0.359·31-s + 1.41·32-s − 1.02·34-s − 1.31·37-s + 0.324·38-s + 1.24·41-s + 0.152·43-s + 0.904·44-s − 1.17·46-s + 0.437·47-s + 2/7·49-s + 1.66·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9571149264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9571149264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.606183820437833600103441591404, −7.894260554784329962490563183694, −6.88825456221220294222029117452, −6.53510001975850170227378349519, −5.79207551816884332992761614767, −4.51278071469945430408773548975, −3.62131961487037887495504734941, −2.83322239075698896831855159072, −1.44946100306976919245519280375, −0.76619087385360988407104513132,
0.76619087385360988407104513132, 1.44946100306976919245519280375, 2.83322239075698896831855159072, 3.62131961487037887495504734941, 4.51278071469945430408773548975, 5.79207551816884332992761614767, 6.53510001975850170227378349519, 6.88825456221220294222029117452, 7.894260554784329962490563183694, 8.606183820437833600103441591404
