L(s) = 1 | − 2.41i·3-s − 2.82·9-s + 1.82·11-s − 6.41i·13-s + 3.58i·17-s − 7.65·19-s + 3.41i·23-s − 0.414i·27-s + 4.65·29-s − 7.41·31-s − 4.41i·33-s + 0.585i·37-s − 15.4·39-s − 3.41·41-s + 0.343i·43-s + ⋯ |
L(s) = 1 | − 1.39i·3-s − 0.942·9-s + 0.551·11-s − 1.77i·13-s + 0.869i·17-s − 1.75·19-s + 0.711i·23-s − 0.0797i·27-s + 0.864·29-s − 1.33·31-s − 0.768i·33-s + 0.0963i·37-s − 2.47·39-s − 0.533·41-s + 0.0523i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4194975020\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4194975020\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2.41iT - 3T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 + 6.41iT - 13T^{2} \) |
| 17 | \( 1 - 3.58iT - 17T^{2} \) |
| 19 | \( 1 + 7.65T + 19T^{2} \) |
| 23 | \( 1 - 3.41iT - 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 - 0.585iT - 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 0.343iT - 43T^{2} \) |
| 47 | \( 1 - 10.8iT - 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 0.585T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.07iT - 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 9.72iT - 97T^{2} \) |
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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
−7.83647819005030714346928609570, −7.08107250260714001579473519490, −6.28034384237319211305220564615, −5.95310373755072114544475806772, −4.94278511029471340614943558275, −3.87969631029833359982500057698, −3.01528796512336504031725523650, −2.02285895916795555141942291371, −1.27283525031928293119680788875, −0.11002770917090099330933671238,
1.69411049503441972205738010280, 2.67837764968245256803879699087, 3.79988508420969494753271754208, 4.31292427244767476114838230713, 4.76461392640088240138632872923, 5.75210722144876760434859230952, 6.66097924288702800192931679815, 7.07671778387897068533933246152, 8.351690245971604146324054438246, 9.044786909556094795935054420862