L(s) = 1 | + 1.27·2-s − 0.380·4-s + 0.255·7-s − 3.02·8-s − 4.63·11-s + 5.02·13-s + 0.325·14-s − 3.09·16-s − 0.336·17-s − 2.91·19-s − 5.89·22-s + 8.65·23-s + 6.39·26-s − 0.0973·28-s − 29-s − 3.26·31-s + 2.12·32-s − 0.428·34-s − 3.86·37-s − 3.70·38-s − 5.71·41-s + 6.98·43-s + 1.76·44-s + 11.0·46-s − 0.336·47-s − 6.93·49-s − 1.91·52-s + ⋯ |
L(s) = 1 | + 0.899·2-s − 0.190·4-s + 0.0966·7-s − 1.07·8-s − 1.39·11-s + 1.39·13-s + 0.0870·14-s − 0.773·16-s − 0.0815·17-s − 0.668·19-s − 1.25·22-s + 1.80·23-s + 1.25·26-s − 0.0183·28-s − 0.185·29-s − 0.587·31-s + 0.374·32-s − 0.0734·34-s − 0.636·37-s − 0.601·38-s − 0.892·41-s + 1.06·43-s + 0.265·44-s + 1.62·46-s − 0.0490·47-s − 0.990·49-s − 0.265·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.251854505\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.251854505\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 7 | \( 1 - 0.255T + 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 0.336T + 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 - 8.65T + 23T^{2} \) |
| 31 | \( 1 + 3.26T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 + 5.71T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 + 0.336T + 47T^{2} \) |
| 53 | \( 1 - 6.01T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 8.46T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 7.60T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.079872230785240634969895088954, −7.17642726989049656810272997501, −6.42117848632088305554292098144, −5.67343793182692968986474669425, −5.14328665978477789879871001250, −4.49828547030636317252325901144, −3.58196850224209353324566565348, −3.06179591198553930768866513235, −2.05229959975341366706974707266, −0.66468111991746026606680549991,
0.66468111991746026606680549991, 2.05229959975341366706974707266, 3.06179591198553930768866513235, 3.58196850224209353324566565348, 4.49828547030636317252325901144, 5.14328665978477789879871001250, 5.67343793182692968986474669425, 6.42117848632088305554292098144, 7.17642726989049656810272997501, 8.079872230785240634969895088954