L(s) = 1 | + (−1.87 − 1.87i)3-s + (−1.41 − 1.41i)7-s + 4i·9-s + 5.29i·11-s + (−1.87 − 1.87i)13-s − 5.29·19-s + 5.29i·21-s + (4.69 − 0.957i)23-s + (1.87 − 1.87i)27-s − i·29-s + 3·31-s + (9.89 − 9.89i)33-s + (7.07 + 7.07i)37-s + 7i·39-s + 9·41-s + ⋯ |
L(s) = 1 | + (−1.08 − 1.08i)3-s + (−0.534 − 0.534i)7-s + 1.33i·9-s + 1.59i·11-s + (−0.518 − 0.518i)13-s − 1.21·19-s + 1.15i·21-s + (0.979 − 0.199i)23-s + (0.360 − 0.360i)27-s − 0.185i·29-s + 0.538·31-s + (1.72 − 1.72i)33-s + (1.16 + 1.16i)37-s + 1.12i·39-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8476136626\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8476136626\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-4.69 + 0.957i)T \) |
good | 3 | \( 1 + (1.87 + 1.87i)T + 3iT^{2} \) |
| 7 | \( 1 + (1.41 + 1.41i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (1.87 + 1.87i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 29 | \( 1 + iT - 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (-7.07 - 7.07i)T + 37iT^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + (4.24 - 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.87 + 1.87i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.82 - 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 5.29iT - 61T^{2} \) |
| 67 | \( 1 + (-1.41 - 1.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (9.35 + 9.35i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + (-1.41 - 1.41i)T + 97iT^{2} \) |
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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
−8.886327069676780124034298542942, −7.69039104880065610115865443027, −7.33631667642556022184530909325, −6.53871998801030378881950404141, −6.08003013252388903971895007033, −4.90425442288208628700275180457, −4.37298716988793184415315362739, −2.85155885850492773733274110577, −1.80862356821508898085653093655, −0.63301123317797013825999648810,
0.59981118034973756793514344471, 2.50793221950249397180888325910, 3.53001105926780374002526551296, 4.38680513420255964593624739195, 5.17343182487894650740950348078, 6.05884225777481952462144432913, 6.28400641808585063759284564977, 7.51070738387869102395755368757, 8.689126649441828849769734238258, 9.135559483609130051775016867898