L(s) = 1 | + i·2-s − 4-s − i·8-s − 11-s + 6i·13-s + 16-s + 2i·17-s + 4·19-s − i·22-s − 6·26-s − 10·29-s + i·32-s − 2·34-s − 6i·37-s + 4i·38-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.353i·8-s − 0.301·11-s + 1.66i·13-s + 0.250·16-s + 0.485i·17-s + 0.917·19-s − 0.213i·22-s − 1.17·26-s − 1.85·29-s + 0.176i·32-s − 0.342·34-s − 0.986i·37-s + 0.648i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6857343642\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6857343642\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−8.670563762419705587139263403963, −7.77173886092578000657023967578, −7.23663214637682138777299255441, −6.59811478441517563200216281847, −5.75641638655036865906530279548, −5.18632533652148509120793466199, −4.18897595931298884188577851589, −3.69234893575818477385860524501, −2.40129407964185499217871152780, −1.40147386139176919626478970156,
0.19308061830019867957196254470, 1.27350617475979543111695547732, 2.44455977518418987463763315738, 3.19597223770976866348755001487, 3.83156702845187141458932196008, 5.07103594796364094799336229336, 5.35124486263614692992689111903, 6.26745913427216178063214876511, 7.41557009097787577091059953207, 7.80621756321457333464519571698