| L(s) = 1 | + 2.18·2-s − 6.08·3-s − 3.21·4-s − 13.3·6-s − 10.8·7-s − 24.5·8-s + 10.0·9-s + 0.656·11-s + 19.5·12-s − 48.6·13-s − 23.7·14-s − 27.8·16-s − 17·17-s + 21.9·18-s + 118.·19-s + 65.9·21-s + 1.43·22-s − 84.7·23-s + 149.·24-s − 106.·26-s + 103.·27-s + 34.8·28-s + 84.8·29-s − 264.·31-s + 135.·32-s − 3.99·33-s − 37.1·34-s + ⋯ |
| L(s) = 1 | + 0.773·2-s − 1.17·3-s − 0.402·4-s − 0.905·6-s − 0.585·7-s − 1.08·8-s + 0.372·9-s + 0.0180·11-s + 0.471·12-s − 1.03·13-s − 0.452·14-s − 0.435·16-s − 0.242·17-s + 0.287·18-s + 1.43·19-s + 0.685·21-s + 0.0139·22-s − 0.768·23-s + 1.27·24-s − 0.802·26-s + 0.735·27-s + 0.235·28-s + 0.543·29-s − 1.53·31-s + 0.747·32-s − 0.0210·33-s − 0.187·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8781160301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8781160301\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 - 2.18T + 8T^{2} \) |
| 3 | \( 1 + 6.08T + 27T^{2} \) |
| 7 | \( 1 + 10.8T + 343T^{2} \) |
| 11 | \( 1 - 0.656T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.6T + 2.19e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 84.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 381.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 32.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 90.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 46.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 385.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 354.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 105.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 330.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 243.T + 9.12e5T^{2} \) |
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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
−11.01818858458137453582322230428, −9.826311014041733814619821098303, −9.265539916003096577184321520949, −7.79882104380231396311583700302, −6.63879294861329780509197738215, −5.74114083730948989949052760919, −5.11415209309490448331125726191, −4.07560787337388165712347489896, −2.79603489401139016011938930327, −0.55102546575634318945206116300,
0.55102546575634318945206116300, 2.79603489401139016011938930327, 4.07560787337388165712347489896, 5.11415209309490448331125726191, 5.74114083730948989949052760919, 6.63879294861329780509197738215, 7.79882104380231396311583700302, 9.265539916003096577184321520949, 9.826311014041733814619821098303, 11.01818858458137453582322230428
