L(s) = 1 | − 1.07·3-s − 3.26i·5-s + i·7-s − 1.84·9-s − i·11-s + (1.30 − 3.36i)13-s + 3.50i·15-s + 3.05·17-s + 3.15i·19-s − 1.07i·21-s + 5.49·23-s − 5.63·25-s + 5.20·27-s + 4.12·29-s − 6.96i·31-s + ⋯ |
L(s) = 1 | − 0.621·3-s − 1.45i·5-s + 0.377i·7-s − 0.614·9-s − 0.301i·11-s + (0.361 − 0.932i)13-s + 0.905i·15-s + 0.740·17-s + 0.724i·19-s − 0.234i·21-s + 1.14·23-s − 1.12·25-s + 1.00·27-s + 0.766·29-s − 1.25i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.320763303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320763303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-1.30 + 3.36i)T \) |
good | 3 | \( 1 + 1.07T + 3T^{2} \) |
| 5 | \( 1 + 3.26iT - 5T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 - 3.15iT - 19T^{2} \) |
| 23 | \( 1 - 5.49T + 23T^{2} \) |
| 29 | \( 1 - 4.12T + 29T^{2} \) |
| 31 | \( 1 + 6.96iT - 31T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 - 0.829iT - 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 9.75iT - 47T^{2} \) |
| 53 | \( 1 - 7.58T + 53T^{2} \) |
| 59 | \( 1 + 15.0iT - 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 - 9.94iT - 67T^{2} \) |
| 71 | \( 1 + 14.2iT - 71T^{2} \) |
| 73 | \( 1 - 7.06iT - 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 0.295iT - 83T^{2} \) |
| 89 | \( 1 - 8.17iT - 89T^{2} \) |
| 97 | \( 1 + 9.48iT - 97T^{2} \) |
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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
−8.265928503372400476162704077973, −7.78445485907508464975755910454, −6.49051672176223964527780720876, −5.78940122751823949095969023772, −5.30850304632614661560052681985, −4.72737509458732552191544390605, −3.61696793270652364710984626711, −2.70137019267261210021494513803, −1.26665997357694286251826888908, −0.52766596524892304750769902346,
1.06497103971239074793356454567, 2.51777834096143693378849808888, 3.10492560513614944198086989533, 4.09050637990791495997131913298, 4.98464463077447387485997512867, 5.89843580956891437007583129362, 6.49957544571526193354266123111, 7.14054689949637653471933057693, 7.61002762907543996522168723157, 8.842709618172817386223949210915