L(s) = 1 | + i·2-s − 4-s + (0.707 + 2.12i)5-s + 0.414i·7-s − i·8-s + (−2.12 + 0.707i)10-s + 3.58·11-s + 3.24i·13-s − 0.414·14-s + 16-s − 6.41i·17-s + 7.82·19-s + (−0.707 − 2.12i)20-s + 3.58i·22-s + 3i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.316 + 0.948i)5-s + 0.156i·7-s − 0.353i·8-s + (−0.670 + 0.223i)10-s + 1.08·11-s + 0.899i·13-s − 0.110·14-s + 0.250·16-s − 1.55i·17-s + 1.79·19-s + (−0.158 − 0.474i)20-s + 0.764i·22-s + 0.625i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.126306092\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.126306092\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 2.12i)T \) |
| 37 | \( 1 + iT \) |
good | 7 | \( 1 - 0.414iT - 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 13 | \( 1 - 3.24iT - 13T^{2} \) |
| 17 | \( 1 + 6.41iT - 17T^{2} \) |
| 19 | \( 1 - 7.82T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 41 | \( 1 - 9.41T + 41T^{2} \) |
| 43 | \( 1 - 6.48iT - 43T^{2} \) |
| 47 | \( 1 + 7.89iT - 47T^{2} \) |
| 53 | \( 1 - 1.82iT - 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.82iT - 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 7.82iT - 73T^{2} \) |
| 79 | \( 1 + 3.75T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 18.7iT - 97T^{2} \) |
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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
−9.044085013236539428392505381180, −7.84069419316338519530388621522, −7.16319785722791459015456354103, −6.81695663087047428760103504086, −5.89415598897672982755498625803, −5.27848962296933944212535159378, −4.22543932792871929260577073227, −3.39963307805160160114423976814, −2.45364399517417283765646056628, −1.11730328185203863398007475186,
0.821658732579419575825030582974, 1.50116643417138729373587037472, 2.69457423616743438358077326095, 3.80906244822043070276087137758, 4.28432362210170583100084624115, 5.43894199230826200695674049974, 5.81929457276584053449617401901, 6.91884896837383320930239674332, 7.996214818308718025897825499016, 8.422273481806888590390445428081