L(s) = 1 | − 3.28·3-s + 1.93·7-s + 7.80·9-s + 5.62·11-s + 2.07·13-s − 3.42·17-s + 19-s − 6.35·21-s + 5.35·23-s − 15.7·27-s + 1.09·29-s + 3.09·31-s − 18.5·33-s − 3.28·37-s − 6.80·39-s − 11.6·41-s + 0.501·43-s + 12.6·47-s − 3.26·49-s + 11.2·51-s − 3.01·53-s − 3.28·57-s − 2.35·59-s + 7.62·61-s + 15.0·63-s − 12.2·67-s − 17.6·69-s + ⋯ |
L(s) = 1 | − 1.89·3-s + 0.730·7-s + 2.60·9-s + 1.69·11-s + 0.574·13-s − 0.830·17-s + 0.229·19-s − 1.38·21-s + 1.11·23-s − 3.03·27-s + 0.202·29-s + 0.555·31-s − 3.22·33-s − 0.540·37-s − 1.08·39-s − 1.81·41-s + 0.0764·43-s + 1.84·47-s − 0.466·49-s + 1.57·51-s − 0.413·53-s − 0.435·57-s − 0.306·59-s + 0.976·61-s + 1.89·63-s − 1.49·67-s − 2.12·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.169968785\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.169968785\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 3.28T + 3T^{2} \) |
| 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 + 3.42T + 17T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 0.501T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 5.35T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 + 3.01T + 97T^{2} \) |
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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
−9.281796194264868449493994352445, −8.532124547447742290563363684523, −7.25463670488283591841824228027, −6.66692575852859823503223693550, −6.11506100451972690296261397757, −5.14406038875529619707710151653, −4.55157960116309887987025672363, −3.69879517333681734992166708263, −1.70233081871925013059287140277, −0.880951095602506269429996086031,
0.880951095602506269429996086031, 1.70233081871925013059287140277, 3.69879517333681734992166708263, 4.55157960116309887987025672363, 5.14406038875529619707710151653, 6.11506100451972690296261397757, 6.66692575852859823503223693550, 7.25463670488283591841824228027, 8.532124547447742290563363684523, 9.281796194264868449493994352445