| L(s) = 1 | + (−1.79 − 1.32i)5-s + 0.476i·7-s − 3.39·11-s − 2.28i·13-s − 2.61i·17-s − 5.08·19-s + i·23-s + (1.47 + 4.77i)25-s − 3.19·29-s + 3.65·31-s + (0.632 − 0.857i)35-s + 8.44i·37-s − 5.25·41-s + 0.269i·43-s − 8.26i·47-s + ⋯ |
| L(s) = 1 | + (−0.804 − 0.593i)5-s + 0.180i·7-s − 1.02·11-s − 0.634i·13-s − 0.634i·17-s − 1.16·19-s + 0.208i·23-s + (0.295 + 0.955i)25-s − 0.592·29-s + 0.655·31-s + (0.106 − 0.145i)35-s + 1.38i·37-s − 0.820·41-s + 0.0410i·43-s − 1.20i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8071712631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8071712631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.79 + 1.32i)T \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 - 0.476iT - 7T^{2} \) |
| 11 | \( 1 + 3.39T + 11T^{2} \) |
| 13 | \( 1 + 2.28iT - 13T^{2} \) |
| 17 | \( 1 + 2.61iT - 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 - 8.44iT - 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 0.269iT - 43T^{2} \) |
| 47 | \( 1 + 8.26iT - 47T^{2} \) |
| 53 | \( 1 - 7.77iT - 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 - 5.45T + 61T^{2} \) |
| 67 | \( 1 + 6.83iT - 67T^{2} \) |
| 71 | \( 1 + 6.28T + 71T^{2} \) |
| 73 | \( 1 - 4.90iT - 73T^{2} \) |
| 79 | \( 1 - 9.29T + 79T^{2} \) |
| 83 | \( 1 + 6.19iT - 83T^{2} \) |
| 89 | \( 1 + 0.423T + 89T^{2} \) |
| 97 | \( 1 - 8.67iT - 97T^{2} \) |
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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
−8.458942658409271495516930030075, −7.88333836219916053697401974841, −7.23657949639143947628427458634, −6.33355233737523634511533439592, −5.33617440034480285915478652312, −4.90024806157015608038666440950, −3.98122312856823406848073016917, −3.10756195091028731338168955418, −2.18940939791622944136944065656, −0.78069405463034485309324069745,
0.31281668061671267295653460601, 1.97514388169613764405132064046, 2.78146557675333073287465271705, 3.84007525527954927740538645411, 4.32256645928829321364529166014, 5.32579372541192491807299440589, 6.23279504078321777878176508758, 6.91509460360579133248427759038, 7.57660512165354677481152183396, 8.277312840159466886570880246403
