L(s) = 1 | + (0.896 − 1.76i)3-s + (−0.951 + 0.309i)4-s − i·5-s + (−1.70 − 2.34i)9-s + (−0.309 + 1.95i)12-s + (−1.76 − 0.896i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)20-s + (−0.142 − 0.896i)23-s − 25-s + (−3.71 + 0.587i)27-s + (−0.363 + 1.11i)31-s + (2.34 + 1.70i)36-s + (0.221 − 1.39i)37-s + (−2.34 + 1.70i)45-s + ⋯ |
L(s) = 1 | + (0.896 − 1.76i)3-s + (−0.951 + 0.309i)4-s − i·5-s + (−1.70 − 2.34i)9-s + (−0.309 + 1.95i)12-s + (−1.76 − 0.896i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)20-s + (−0.142 − 0.896i)23-s − 25-s + (−3.71 + 0.587i)27-s + (−0.363 + 1.11i)31-s + (2.34 + 1.70i)36-s + (0.221 − 1.39i)37-s + (−2.34 + 1.70i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.066525813\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066525813\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 3 | \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.221 + 1.39i)T + (-0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \) |
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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
−8.562018981729626721246072335678, −7.75294577596072928719415700749, −7.40716938110407582557546026770, −6.29129983658081058815754957802, −5.59788104199579749024713006904, −4.52903096233061350926861528611, −3.63438905361237460547321576734, −2.67795644191708803768695534536, −1.60434644537388216007708758558, −0.60685046905663235015073312484,
2.18312043743573324161824189927, 3.18243370903278324474388745145, 3.81010428072835754295885415363, 4.41360954309058914768312082272, 5.33737464939892339591455842083, 5.89796913694760574613871589787, 7.26018365617317004671303747689, 8.135538501130149320843979931667, 8.664165352304605300633342674779, 9.479371676403777641388380100800