L(s) = 1 | + 2.52i·2-s + 0.473i·3-s − 4.38·4-s + (2.21 + 0.267i)5-s − 1.19·6-s − 1.17i·7-s − 6.03i·8-s + 2.77·9-s + (−0.677 + 5.61i)10-s + 11-s − 2.07i·12-s + 2.47i·13-s + 2.97·14-s + (−0.126 + 1.05i)15-s + 6.46·16-s − 1.14i·17-s + ⋯ |
L(s) = 1 | + 1.78i·2-s + 0.273i·3-s − 2.19·4-s + (0.992 + 0.119i)5-s − 0.488·6-s − 0.444i·7-s − 2.13i·8-s + 0.925·9-s + (−0.214 + 1.77i)10-s + 0.301·11-s − 0.599i·12-s + 0.685i·13-s + 0.793·14-s + (−0.0327 + 0.271i)15-s + 1.61·16-s − 0.278i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.779078845\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.779078845\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.21 - 0.267i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 - 2.52iT - 2T^{2} \) |
| 3 | \( 1 - 0.473iT - 3T^{2} \) |
| 7 | \( 1 + 1.17iT - 7T^{2} \) |
| 13 | \( 1 - 2.47iT - 13T^{2} \) |
| 17 | \( 1 + 1.14iT - 17T^{2} \) |
| 23 | \( 1 - 7.90iT - 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 - 9.43iT - 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 - 5.71iT - 43T^{2} \) |
| 47 | \( 1 + 5.71iT - 47T^{2} \) |
| 53 | \( 1 + 6.53iT - 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 1.57T + 61T^{2} \) |
| 67 | \( 1 - 1.34iT - 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 + 13.9iT - 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 + 1.27iT - 83T^{2} \) |
| 89 | \( 1 + 0.835T + 89T^{2} \) |
| 97 | \( 1 - 1.40iT - 97T^{2} \) |
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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
−9.715337133598972408811474375661, −9.537500971663456082501961193359, −8.577480262530050045881719745550, −7.51554850737779842503487351125, −6.90363386834866204100917058576, −6.28895381874228693176026967836, −5.26684304238891316247623336396, −4.63882012977773747177416358119, −3.54808828745555504057348443313, −1.50404188705024698592105471656,
0.921257103962304226282214713117, 1.97696479251698416763925092951, 2.74101360364566635725211315475, 3.96664235792709031403450058929, 4.89092453001541471810694648761, 5.85764414796686081228405695793, 6.96287926102461043758511290850, 8.311924884582420085665539479498, 9.087651931561837660181923513378, 9.695684742766311687989435362361