| L(s) = 1 | − 5.12·3-s − 20.1·5-s + 21.0·7-s − 0.692·9-s + 17.0·11-s − 78.4·13-s + 103.·15-s − 17·17-s + 139.·19-s − 108.·21-s − 138.·23-s + 279.·25-s + 142.·27-s + 203.·29-s − 42.6·31-s − 87.6·33-s − 423.·35-s + 39.9·37-s + 402.·39-s + 313.·41-s − 229.·43-s + 13.9·45-s + 322.·47-s + 101.·49-s + 87.1·51-s + 213.·53-s − 343.·55-s + ⋯ |
| L(s) = 1 | − 0.987·3-s − 1.79·5-s + 1.13·7-s − 0.0256·9-s + 0.468·11-s − 1.67·13-s + 1.77·15-s − 0.242·17-s + 1.68·19-s − 1.12·21-s − 1.25·23-s + 2.23·25-s + 1.01·27-s + 1.30·29-s − 0.247·31-s − 0.462·33-s − 2.04·35-s + 0.177·37-s + 1.65·39-s + 1.19·41-s − 0.813·43-s + 0.0461·45-s + 0.999·47-s + 0.294·49-s + 0.239·51-s + 0.553·53-s − 0.842·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 5.12T + 27T^{2} \) |
| 5 | \( 1 + 20.1T + 125T^{2} \) |
| 7 | \( 1 - 21.0T + 343T^{2} \) |
| 11 | \( 1 - 17.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 42.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 39.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 229.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 322.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 213.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 488.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 382.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 426.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 483.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 637.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 120.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 465.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−8.955489926188739976788722224264, −7.936623275652705969704407298408, −7.58175535073487922594470080071, −6.68928609469447014176023603495, −5.37939156486724509255395433149, −4.76286423483634184723785194342, −4.02371873605736981321466511462, −2.71916288528587667658767294716, −0.997036942324322170081590255520, 0,
0.997036942324322170081590255520, 2.71916288528587667658767294716, 4.02371873605736981321466511462, 4.76286423483634184723785194342, 5.37939156486724509255395433149, 6.68928609469447014176023603495, 7.58175535073487922594470080071, 7.936623275652705969704407298408, 8.955489926188739976788722224264
