L(s) = 1 | + (0.366 + 0.366i)2-s − 0.732i·4-s + (−0.707 + 0.707i)7-s + (0.633 − 0.633i)8-s − 1.93i·11-s − 0.517·14-s − 0.267·16-s + (0.707 − 0.707i)22-s + (1.36 − 1.36i)23-s + (0.517 + 0.517i)28-s + 0.517·29-s + (−0.732 − 0.732i)32-s + (−0.707 + 0.707i)37-s + (0.707 + 0.707i)43-s − 1.41·44-s + ⋯ |
L(s) = 1 | + (0.366 + 0.366i)2-s − 0.732i·4-s + (−0.707 + 0.707i)7-s + (0.633 − 0.633i)8-s − 1.93i·11-s − 0.517·14-s − 0.267·16-s + (0.707 − 0.707i)22-s + (1.36 − 1.36i)23-s + (0.517 + 0.517i)28-s + 0.517·29-s + (−0.732 − 0.732i)32-s + (−0.707 + 0.707i)37-s + (0.707 + 0.707i)43-s − 1.41·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.236888347\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.236888347\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 11 | \( 1 + 1.93iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 29 | \( 1 - 0.517T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 71 | \( 1 + 0.517iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.333108473931088115774384910537, −8.848485231499468168517041575977, −7.980734504600801961077484527043, −6.63028795970419229790062518053, −6.33664940427641948292983789309, −5.50947716002468029369939036991, −4.76124582424633634832530299355, −3.48512638490808730413728705295, −2.63269229092988112320683725222, −0.921860638322947005572199690746,
1.74940398504564107309680385852, 2.88680937668118379592671039789, 3.79161089902089641625625424940, 4.52455558546362392579530879157, 5.37624061816045063627250995126, 6.95391048302716835516072821473, 7.09950950286914234861124215572, 7.970205960913710794190104157396, 9.100008071857990013277522640747, 9.733504672347671665072905386892