| L(s) = 1 | − 10.2i·2-s + 407.·4-s + 625·5-s + (−6.35e3 − 103. i)7-s − 9.38e3i·8-s − 6.37e3i·10-s + 3.22e4i·11-s + 8.60e4i·13-s + (−1.06e3 + 6.48e4i)14-s + 1.12e5·16-s + 1.43e4·17-s − 1.89e5i·19-s + 2.54e5·20-s + 3.29e5·22-s − 5.82e5i·23-s + ⋯ |
| L(s) = 1 | − 0.451i·2-s + 0.796·4-s + 0.447·5-s + (−0.999 − 0.0163i)7-s − 0.810i·8-s − 0.201i·10-s + 0.665i·11-s + 0.835i·13-s + (−0.00737 + 0.451i)14-s + 0.430·16-s + 0.0415·17-s − 0.334i·19-s + 0.356·20-s + 0.300·22-s − 0.433i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6330037203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6330037203\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 + (6.35e3 + 103. i)T \) |
| good | 2 | \( 1 + 10.2iT - 512T^{2} \) |
| 11 | \( 1 - 3.22e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 8.60e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 1.43e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.89e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + 5.82e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 5.36e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 7.52e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 1.95e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.23e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.30e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.49e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.02e6iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 6.58e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.83e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 1.44e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 8.85e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 - 2.24e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 8.97e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.61e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.09e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.98e7iT - 7.60e17T^{2} \) |
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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
−10.31154805259827144155676120010, −9.748049062959511276087869234277, −8.799224900658293753791874654514, −7.23408072821207522649738520209, −6.71486062792806669902208336948, −5.75484477363596413161859672056, −4.31700304951097708354172030324, −3.16338006505104848011558424396, −2.26665976963257994117132316743, −1.30629731636653329943209229737,
0.10471853590096997016870828664, 1.46391156797036519946732682541, 2.73010133464088960875989942505, 3.47681861682071425794492097278, 5.24976900165472995158969775315, 6.04662411234854492853661941134, 6.72702986638833352948209535572, 7.79010700834811811049428085279, 8.709655927996797144478215637882, 9.923125643522682322861349727697
