L(s) = 1 | − 5.44i·11-s − 1.89·17-s + 8.34i·19-s + 5·25-s + 12.7·41-s − 2.34i·43-s − 7·49-s + 9.24i·59-s − 14.3i·67-s + 13.6·73-s − 18i·83-s + 18·89-s − 19.6·97-s − 20.1i·107-s + 18·113-s + ⋯ |
L(s) = 1 | − 1.64i·11-s − 0.460·17-s + 1.91i·19-s + 25-s + 1.99·41-s − 0.358i·43-s − 49-s + 1.20i·59-s − 1.75i·67-s + 1.60·73-s − 1.97i·83-s + 1.90·89-s − 1.99·97-s − 1.94i·107-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.783369023\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783369023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.44iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 8.34iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 + 2.34iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 9.24iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14.3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 18iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + 19.6T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.118669211666567869807373020257, −7.57617664240460767959148250474, −6.49329812885445720164641128735, −6.00848031353517051189339274357, −5.35363084512029727791777773443, −4.34319022650847121626411629696, −3.55021049018825957357041374232, −2.87229226565663761254900900357, −1.69732243938411828731712592776, −0.60163146944137406734504189191,
0.896283787191211981748658947747, 2.17650150114013146620473439577, 2.74782776908733793646077405971, 3.96741569729646861586675690644, 4.73905567477925792688653081204, 5.12003159711086474126154170849, 6.36240660226653678011849186272, 6.89885689313808500797471973479, 7.45310473740587060909484833208, 8.276357428786396083141754412864