| L(s) = 1 | − 1.41·5-s + (−2.44 − i)7-s − 3.46·11-s + 4.24·13-s − 4.89i·17-s + 4.24i·19-s − 2.99·25-s − 4.89·31-s + (3.46 + 1.41i)35-s − 6.92i·37-s + 9.79i·41-s + 3.46·43-s − 4.89·47-s + (4.99 + 4.89i)49-s − 13.8i·53-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + (−0.925 − 0.377i)7-s − 1.04·11-s + 1.17·13-s − 1.18i·17-s + 0.973i·19-s − 0.599·25-s − 0.879·31-s + (0.585 + 0.239i)35-s − 1.13i·37-s + 1.53i·41-s + 0.528·43-s − 0.714·47-s + (0.714 + 0.699i)49-s − 1.90i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8677014301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8677014301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.44 + i)T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4.89T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 9.79iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 13.8iT - 53T^{2} \) |
| 59 | \( 1 - 7.07iT - 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 - 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−8.428403790301153802236310910551, −7.76485786198814523995303614633, −7.22789675991278232619551979852, −6.31577844608654645740213192726, −5.67205168971647268989347650277, −4.76176249062719954421888058924, −3.72384269483551038703734757468, −3.34430355010386565986538541540, −2.21142405776857027013473220163, −0.75514871342674771733566827741,
0.35494873524933593420990089930, 1.87428668967976392852758189548, 2.98295262326766249944305955094, 3.62700320886724357126598402831, 4.43452130626710399318383322228, 5.53001186095651293853678365277, 6.06088264069535320885657576115, 6.87321358253054629315191872500, 7.65619274740010870127904954657, 8.387147085175010203780369748617
