| L(s) = 1 | + 0.631·2-s + 9.24·3-s − 7.60·4-s + 19.0·5-s + 5.83·6-s + 25.4·7-s − 9.84·8-s + 58.4·9-s + 12.0·10-s − 11·11-s − 70.2·12-s + 16.0·14-s + 176.·15-s + 54.5·16-s + 31.0·17-s + 36.8·18-s − 73.2·19-s − 144.·20-s + 235.·21-s − 6.94·22-s − 95.8·23-s − 91.0·24-s + 237.·25-s + 290.·27-s − 193.·28-s + 227.·29-s + 111.·30-s + ⋯ |
| L(s) = 1 | + 0.223·2-s + 1.77·3-s − 0.950·4-s + 1.70·5-s + 0.396·6-s + 1.37·7-s − 0.435·8-s + 2.16·9-s + 0.380·10-s − 0.301·11-s − 1.69·12-s + 0.306·14-s + 3.03·15-s + 0.853·16-s + 0.443·17-s + 0.482·18-s − 0.884·19-s − 1.61·20-s + 2.44·21-s − 0.0672·22-s − 0.868·23-s − 0.774·24-s + 1.90·25-s + 2.07·27-s − 1.30·28-s + 1.45·29-s + 0.676·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.836879367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.836879367\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.631T + 8T^{2} \) |
| 3 | \( 1 - 9.24T + 27T^{2} \) |
| 5 | \( 1 - 19.0T + 125T^{2} \) |
| 7 | \( 1 - 25.4T + 343T^{2} \) |
| 17 | \( 1 - 31.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 73.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 95.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 227.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 5.28T + 2.97e4T^{2} \) |
| 37 | \( 1 + 304.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 172.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 411.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 185.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 703.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 519.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 669.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 42.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 57.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 521.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.69e3T + 9.12e5T^{2} \) |
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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
−8.742429662441622196798782970221, −8.388988100778303649391240361968, −7.67775503598720319723585844340, −6.47098040199615001553658835884, −5.42087097158007325860189689248, −4.73921377007137518798262224112, −3.89507721758285843407195862468, −2.74358953395400774812304785736, −2.02087636543547109533500457193, −1.23933955005970867389491257830,
1.23933955005970867389491257830, 2.02087636543547109533500457193, 2.74358953395400774812304785736, 3.89507721758285843407195862468, 4.73921377007137518798262224112, 5.42087097158007325860189689248, 6.47098040199615001553658835884, 7.67775503598720319723585844340, 8.388988100778303649391240361968, 8.742429662441622196798782970221
