| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 20.1·7-s + 8·8-s + 9·9-s − 10·10-s − 22.4·11-s + 12·12-s + 11.0·13-s + 40.2·14-s − 15·15-s + 16·16-s + 127.·17-s + 18·18-s − 33.8·19-s − 20·20-s + 60.3·21-s − 44.8·22-s − 122.·23-s + 24·24-s + 25·25-s + 22.1·26-s + 27·27-s + 80.4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.08·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.614·11-s + 0.288·12-s + 0.236·13-s + 0.768·14-s − 0.258·15-s + 0.250·16-s + 1.81·17-s + 0.235·18-s − 0.408·19-s − 0.223·20-s + 0.627·21-s − 0.434·22-s − 1.10·23-s + 0.204·24-s + 0.200·25-s + 0.167·26-s + 0.192·27-s + 0.543·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.588523298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.588523298\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 31 | \( 1 + 31T \) |
| good | 7 | \( 1 - 20.1T + 343T^{2} \) |
| 11 | \( 1 + 22.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 127.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 276.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 334.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 91.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 279.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 154.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 592.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 631.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 589.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 754.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 935.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 258.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 858.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 393.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
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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
−9.907632306713318635358491599686, −8.431348648795426447474934640114, −8.051703968995523747094209461840, −7.31751836313060055727922066715, −6.09674669152550153467474234580, −5.13647228148159280349402948924, −4.32248534584922923229327986758, −3.36922833406205200266247958877, −2.32725708426474337845316718124, −1.09452316109066989739620887470,
1.09452316109066989739620887470, 2.32725708426474337845316718124, 3.36922833406205200266247958877, 4.32248534584922923229327986758, 5.13647228148159280349402948924, 6.09674669152550153467474234580, 7.31751836313060055727922066715, 8.051703968995523747094209461840, 8.431348648795426447474934640114, 9.907632306713318635358491599686
