L(s) = 1 | − 1.36·2-s − 0.141·4-s − 4.14·7-s + 2.91·8-s + 1.77·11-s − 3·13-s + 5.64·14-s − 3.69·16-s + 3.14·17-s + 2.86·19-s − 2.42·22-s − 6.15·23-s + 4.08·26-s + 0.585·28-s − 1.36·29-s − 5.86·31-s − 0.797·32-s − 4.28·34-s + 7.00·37-s − 3.91·38-s + 3.58·41-s + 43-s − 0.251·44-s + 8.38·46-s + 12.6·47-s + 10.1·49-s + 0.424·52-s + ⋯ |
L(s) = 1 | − 0.964·2-s − 0.0706·4-s − 1.56·7-s + 1.03·8-s + 0.536·11-s − 0.832·13-s + 1.50·14-s − 0.924·16-s + 0.761·17-s + 0.657·19-s − 0.516·22-s − 1.28·23-s + 0.802·26-s + 0.110·28-s − 0.253·29-s − 1.05·31-s − 0.141·32-s − 0.734·34-s + 1.15·37-s − 0.634·38-s + 0.559·41-s + 0.152·43-s − 0.0378·44-s + 1.23·46-s + 1.84·47-s + 1.45·49-s + 0.0587·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 3.14T + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 29 | \( 1 + 1.36T + 29T^{2} \) |
| 31 | \( 1 + 5.86T + 31T^{2} \) |
| 37 | \( 1 - 7.00T + 37T^{2} \) |
| 41 | \( 1 - 3.58T + 41T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.86T + 53T^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 0.282T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 4.55T + 97T^{2} \) |
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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
−7.59766238335122171898835970633, −6.82932042503882766674415601579, −6.06470790640337593356752631267, −5.46885347516392311493639927490, −4.41419160186490764630939119953, −3.78170832289246370215399466452, −2.98834700022929616800229546205, −2.01918716772857776788422049166, −0.896912986169064803367874093887, 0,
0.896912986169064803367874093887, 2.01918716772857776788422049166, 2.98834700022929616800229546205, 3.78170832289246370215399466452, 4.41419160186490764630939119953, 5.46885347516392311493639927490, 6.06470790640337593356752631267, 6.82932042503882766674415601579, 7.59766238335122171898835970633