| L(s) = 1 | + 2.69e3i·3-s − 3.49e4i·5-s + (−8.10e5 − 1.46e5i)7-s − 2.46e6·9-s + 7.58e6·11-s − 1.05e8i·13-s + 9.40e7·15-s + 2.67e8i·17-s + 1.14e9i·19-s + (3.93e8 − 2.18e9i)21-s + 6.66e9·23-s − 1.22e9·25-s + 6.24e9i·27-s − 1.96e10·29-s + 4.35e10i·31-s + ⋯ |
| L(s) = 1 | + 1.23i·3-s − 0.447i·5-s + (−0.984 − 0.177i)7-s − 0.514·9-s + 0.389·11-s − 1.68i·13-s + 0.550·15-s + 0.651i·17-s + 1.27i·19-s + (0.218 − 1.21i)21-s + 1.95·23-s − 0.199·25-s + 0.597i·27-s − 1.13·29-s + 1.58i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.4510152692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4510152692\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.49e4iT \) |
| 7 | \( 1 + (8.10e5 + 1.46e5i)T \) |
| good | 3 | \( 1 - 2.69e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 - 7.58e6T + 3.79e14T^{2} \) |
| 13 | \( 1 + 1.05e8iT - 3.93e15T^{2} \) |
| 17 | \( 1 - 2.67e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.14e9iT - 7.99e17T^{2} \) |
| 23 | \( 1 - 6.66e9T + 1.15e19T^{2} \) |
| 29 | \( 1 + 1.96e10T + 2.97e20T^{2} \) |
| 31 | \( 1 - 4.35e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 9.29e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 2.05e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 1.52e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 1.67e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.44e12T + 1.37e24T^{2} \) |
| 59 | \( 1 - 1.04e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 6.16e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 6.72e11T + 3.67e25T^{2} \) |
| 71 | \( 1 + 5.19e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + 2.54e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 + 2.99e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 2.90e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 5.24e12iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 1.53e14iT - 6.52e27T^{2} \) |
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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
−10.29984681723212080669076841510, −9.398342645554818175739321727144, −8.542534576084761687807224810717, −7.18718377912525467809231705014, −5.80022072854510478345502621038, −4.99561041146410608822069040019, −3.67435788970709507545218623405, −3.22183032964617289150227343677, −1.34191389636027464976868549757, −0.094553813129500616642826243829,
1.00684868032222629694103866659, 2.13576657029661303732250555896, 3.04082014815232780304219964040, 4.45185195601922669298899906578, 6.04661383204936232676235720164, 6.95822624908868092451501234558, 7.23199808455672188649786505922, 8.960907097832875071075423834399, 9.550430302584465682529328236428, 11.19377388633419539832015085185
