L(s) = 1 | + 2-s + (−0.403 − 1.68i)3-s + 4-s + (−0.403 − 1.68i)6-s + (1.28 + 2.31i)7-s + 8-s + (−2.67 + 1.35i)9-s + 5.34i·11-s + (−0.403 − 1.68i)12-s + 3.95·13-s + (1.28 + 2.31i)14-s + 16-s + 7.32i·17-s + (−2.67 + 1.35i)18-s − 0.807i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.232 − 0.972i)3-s + 0.5·4-s + (−0.164 − 0.687i)6-s + (0.484 + 0.874i)7-s + 0.353·8-s + (−0.891 + 0.453i)9-s + 1.61i·11-s + (−0.116 − 0.486i)12-s + 1.09·13-s + (0.342 + 0.618i)14-s + 0.250·16-s + 1.77i·17-s + (−0.630 + 0.320i)18-s − 0.185i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.384705494\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.384705494\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.403 + 1.68i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.28 - 2.31i)T \) |
good | 11 | \( 1 - 5.34iT - 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 - 7.32iT - 17T^{2} \) |
| 19 | \( 1 + 0.807iT - 19T^{2} \) |
| 23 | \( 1 - 0.281T + 23T^{2} \) |
| 29 | \( 1 - 0.281iT - 29T^{2} \) |
| 31 | \( 1 + 9.07iT - 31T^{2} \) |
| 37 | \( 1 + 6.06iT - 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 - 6.34iT - 43T^{2} \) |
| 47 | \( 1 + 5.78iT - 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 - 6.71iT - 67T^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 - 3.26T + 79T^{2} \) |
| 83 | \( 1 - 1.53iT - 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−10.12706388447924291283209314822, −8.892417355233196657452109945989, −8.127418690144326121380880281851, −7.36784527475559053009983491995, −6.34938972104046915492609020797, −5.83248480833648928288677014921, −4.84307709910350991316416719148, −3.77389318149014415181553043308, −2.27481677897418109373972890787, −1.65655707816319752522146437262,
0.946101667346293991653068634513, 3.04613986804789430790926711197, 3.62416730895101115260684975051, 4.66269203589370431526050043848, 5.38715323805954909981284549499, 6.25147333644949419143853997914, 7.20373032264968398518071442328, 8.399341450329244878552826973414, 8.957958994608959958977009909934, 10.23552516392806122412670205137