L(s) = 1 | + (−0.707 − 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + (0.707 − 0.707i)8-s − 9-s + 1.00i·10-s − 1.00·12-s + 13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (0.707 − 0.707i)20-s + (0.707 + 0.707i)24-s + 1.00i·25-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + i·3-s + 1.00i·4-s + (−0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + (0.707 − 0.707i)8-s − 9-s + 1.00i·10-s − 1.00·12-s + 13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (0.707 + 0.707i)18-s + (0.707 − 0.707i)20-s + (0.707 + 0.707i)24-s + 1.00i·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7250815888\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7250815888\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 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
−8.959575649681308491365574519738, −8.358470653430911102815862757443, −7.78818486088298097495705782549, −6.74339155525202695618909673898, −5.60811050288213643409485907942, −4.74689988949350776438952378874, −3.87206190017888613490066255873, −3.51621297133632390362146193382, −2.27885732909406712972943687754, −0.838282418099658473660954116707,
0.870082586072284331922961752546, 2.07999575465429668693267552388, 3.14527855160455437024303906252, 4.25645840718611188655040479630, 5.46934026274358945522738515109, 6.18628122390421954542041919613, 6.79542471563071772365833722645, 7.38011407248040072540568691554, 8.116844839162447758444193624453, 8.531655447206990704210071528805