L(s) = 1 | + 2.37·2-s + 3.65·4-s − 0.726·7-s + 3.92·8-s + 0.273·11-s + 5.95·13-s − 1.72·14-s + 2.02·16-s + 5.27·17-s + 19-s + 0.651·22-s − 3.67·23-s + 14.1·26-s − 2.65·28-s + 2.27·29-s + 3.19·31-s − 3.02·32-s + 12.5·34-s − 8.12·37-s + 2.37·38-s + 9.43·41-s − 9.81·43-s + 44-s − 8.74·46-s + 12.1·47-s − 6.47·49-s + 21.7·52-s + ⋯ |
L(s) = 1 | + 1.68·2-s + 1.82·4-s − 0.274·7-s + 1.38·8-s + 0.0825·11-s + 1.65·13-s − 0.461·14-s + 0.507·16-s + 1.27·17-s + 0.229·19-s + 0.138·22-s − 0.767·23-s + 2.77·26-s − 0.501·28-s + 0.422·29-s + 0.574·31-s − 0.535·32-s + 2.15·34-s − 1.33·37-s + 0.385·38-s + 1.47·41-s − 1.49·43-s + 0.150·44-s − 1.28·46-s + 1.77·47-s − 0.924·49-s + 3.01·52-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
\(L(1)\) |
\(\approx\) |
\(5.866890779\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.866890779\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.37T + 2T^{2} \) |
| 7 | \( 1 + 0.726T + 7T^{2} \) |
| 11 | \( 1 - 0.273T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 - 5.27T + 17T^{2} \) |
| 23 | \( 1 + 3.67T + 23T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 8.12T + 37T^{2} \) |
| 41 | \( 1 - 9.43T + 41T^{2} \) |
| 43 | \( 1 + 9.81T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 5.69T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 + 0.103T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 + 6.67T + 73T^{2} \) |
| 79 | \( 1 - 3.87T + 79T^{2} \) |
| 83 | \( 1 + 0.488T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 4.44T + 97T^{2} \) |
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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
−8.260463990297788191630253105754, −7.40431745638240561064337112510, −6.57783274606217489612406022962, −5.98327309663853387818180422712, −5.48402081525655015474837100765, −4.59537704372274093724485294848, −3.66535353405405435315061388968, −3.40993454429543777336460815027, −2.31121733241419916104656973415, −1.15065948936924680686550817808,
1.15065948936924680686550817808, 2.31121733241419916104656973415, 3.40993454429543777336460815027, 3.66535353405405435315061388968, 4.59537704372274093724485294848, 5.48402081525655015474837100765, 5.98327309663853387818180422712, 6.57783274606217489612406022962, 7.40431745638240561064337112510, 8.260463990297788191630253105754