L(s) = 1 | − i·7-s + 4.24i·11-s + 7i·13-s + 4.24·17-s + 7·19-s − 29.6·23-s + 29.6i·29-s + 17·31-s − 16i·37-s + 50.9i·41-s + 55i·43-s + 46.6·47-s + 48·49-s − 84.8·53-s + 55.1i·59-s + ⋯ |
L(s) = 1 | − 0.142i·7-s + 0.385i·11-s + 0.538i·13-s + 0.249·17-s + 0.368·19-s − 1.29·23-s + 1.02i·29-s + 0.548·31-s − 0.432i·37-s + 1.24i·41-s + 1.27i·43-s + 0.992·47-s + 0.979·49-s − 1.60·53-s + 0.934i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.456736717\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.456736717\) |
\(L(2)\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + iT - 49T^{2} \) |
| 11 | \( 1 - 4.24iT - 121T^{2} \) |
| 13 | \( 1 - 7iT - 169T^{2} \) |
| 17 | \( 1 - 4.24T + 289T^{2} \) |
| 19 | \( 1 - 7T + 361T^{2} \) |
| 23 | \( 1 + 29.6T + 529T^{2} \) |
| 29 | \( 1 - 29.6iT - 841T^{2} \) |
| 31 | \( 1 - 17T + 961T^{2} \) |
| 37 | \( 1 + 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 84.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 55.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 65T + 3.72e3T^{2} \) |
| 67 | \( 1 + 49iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 88iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 40T + 6.24e3T^{2} \) |
| 83 | \( 1 + 156.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 41iT - 9.40e3T^{2} \) |
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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
−10.00124413348300723399448157539, −9.394364014651918409916464397675, −8.379559534149185931994276466278, −7.57733699630330781675164775540, −6.70370347543256590443806310184, −5.79686834783196692910215213436, −4.72648809367660328975493558619, −3.83615956040594936207486671052, −2.59111666076150496945980523348, −1.29324073460812350774489227430,
0.50216710376623250371565499117, 2.08023106676399076684814342289, 3.27384795371068798099968291653, 4.28226274504488356263888716609, 5.48196539488531696122540471028, 6.11648930981321834293180103861, 7.26428124996705531902049974218, 8.069102663774809086281259772689, 8.815926177345634398901748212083, 9.850697554156542018614474822703