| L(s) = 1 | − 1.55e3i·3-s − 3.49e4i·5-s + (−1.88e5 + 8.01e5i)7-s + 2.35e6·9-s − 6.25e6·11-s − 7.13e7i·13-s − 5.44e7·15-s + 1.43e8i·17-s + 1.07e9i·19-s + (1.24e9 + 2.93e8i)21-s − 1.64e9·23-s − 1.22e9·25-s − 1.11e10i·27-s + 6.22e9·29-s − 4.38e10i·31-s + ⋯ |
| L(s) = 1 | − 0.712i·3-s − 0.447i·5-s + (−0.228 + 0.973i)7-s + 0.492·9-s − 0.321·11-s − 1.13i·13-s − 0.318·15-s + 0.348i·17-s + 1.19i·19-s + (0.693 + 0.162i)21-s − 0.483·23-s − 0.199·25-s − 1.06i·27-s + 0.360·29-s − 1.59i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(2.003633778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.003633778\) |
| \(L(8)\) |
|
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 | \( 1 + (1.88e5 - 8.01e5i)T \) |
| good | 3 | \( 1 + 1.55e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 + 6.25e6T + 3.79e14T^{2} \) |
| 13 | \( 1 + 7.13e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 - 1.43e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.07e9iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 1.64e9T + 1.15e19T^{2} \) |
| 29 | \( 1 - 6.22e9T + 2.97e20T^{2} \) |
| 31 | \( 1 + 4.38e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 9.52e10T + 9.01e21T^{2} \) |
| 41 | \( 1 - 1.20e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 3.83e11T + 7.38e22T^{2} \) |
| 47 | \( 1 - 9.30e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 5.02e10T + 1.37e24T^{2} \) |
| 59 | \( 1 - 2.90e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 5.76e11iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 8.65e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 2.78e12T + 8.27e25T^{2} \) |
| 73 | \( 1 - 1.78e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 + 2.59e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 7.49e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 1.45e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 2.16e12iT - 6.52e27T^{2} \) |
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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
−10.42751133762785599620275096055, −9.512210795048008540791377314266, −8.247487512096321482975780928187, −7.66555338506386059920211456188, −6.20242225741763580664433606210, −5.53612579117782254169168374243, −4.13680929660020416225044081792, −2.75200937821939039725055208885, −1.74571091175674364550643244502, −0.69447213257868944882298781550,
0.53803945689753751984842974412, 1.90412330796656914293584588241, 3.29213772022924979998930243274, 4.19619659394717945195556541008, 5.06834993044229577820905498881, 6.75255049190689334222223014330, 7.21348273336224558389381372567, 8.766039025170338387218948026901, 9.778299710497708467607329968852, 10.50882220164052507899884935563
