| L(s) = 1 | + 2·2-s − 0.534·3-s + 4·4-s − 1.06·6-s − 28.9·7-s + 8·8-s − 26.7·9-s + 33.5·11-s − 2.13·12-s + 10.2·13-s − 57.8·14-s + 16·16-s + 56.5·17-s − 53.4·18-s − 5.26·19-s + 15.4·21-s + 67.0·22-s + 23·23-s − 4.27·24-s + 20.5·26-s + 28.7·27-s − 115.·28-s − 162.·29-s − 160.·31-s + 32·32-s − 17.9·33-s + 113.·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.102·3-s + 0.5·4-s − 0.0727·6-s − 1.56·7-s + 0.353·8-s − 0.989·9-s + 0.918·11-s − 0.0514·12-s + 0.219·13-s − 1.10·14-s + 0.250·16-s + 0.807·17-s − 0.699·18-s − 0.0635·19-s + 0.160·21-s + 0.649·22-s + 0.208·23-s − 0.0363·24-s + 0.155·26-s + 0.204·27-s − 0.781·28-s − 1.03·29-s − 0.930·31-s + 0.176·32-s − 0.0945·33-s + 0.570·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.396353580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396353580\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 0.534T + 27T^{2} \) |
| 7 | \( 1 + 28.9T + 343T^{2} \) |
| 11 | \( 1 - 33.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.26T + 6.85e3T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 16.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 333.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 673.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 454.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 132.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 121.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 176.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 563.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 809.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 702.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 342.T + 9.12e5T^{2} \) |
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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
−9.385481698013125165798372273084, −8.793369883529468843265716762498, −7.52364146281734949761004340629, −6.70828142004468152078835060294, −5.96334528608021439875782721295, −5.39476747899534584611149966026, −3.88053683526264648627695816616, −3.41257225760460408810673035941, −2.34436346585886945945887703901, −0.70477841886170280937968777818,
0.70477841886170280937968777818, 2.34436346585886945945887703901, 3.41257225760460408810673035941, 3.88053683526264648627695816616, 5.39476747899534584611149966026, 5.96334528608021439875782721295, 6.70828142004468152078835060294, 7.52364146281734949761004340629, 8.793369883529468843265716762498, 9.385481698013125165798372273084
