| L(s) = 1 | + 1.78i·3-s + (1.46 − 1.69i)5-s − 1.75i·7-s − 0.192·9-s + 4.77·11-s − 1.72i·13-s + (3.02 + 2.61i)15-s − 7.81i·17-s − 2.43·19-s + 3.13·21-s − i·23-s + (−0.731 − 4.94i)25-s + 5.01i·27-s − 7.86·29-s − 6.14·31-s + ⋯ |
| L(s) = 1 | + 1.03i·3-s + (0.653 − 0.757i)5-s − 0.663i·7-s − 0.0640·9-s + 1.43·11-s − 0.479i·13-s + (0.780 + 0.673i)15-s − 1.89i·17-s − 0.559·19-s + 0.684·21-s − 0.208i·23-s + (−0.146 − 0.989i)25-s + 0.965i·27-s − 1.46·29-s − 1.10·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.998754204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998754204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.46 + 1.69i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 1.78iT - 3T^{2} \) |
| 7 | \( 1 + 1.75iT - 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 1.72iT - 13T^{2} \) |
| 17 | \( 1 + 7.81iT - 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 + 6.83iT - 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 + 3.26iT - 43T^{2} \) |
| 47 | \( 1 + 8.46iT - 47T^{2} \) |
| 53 | \( 1 - 2.76iT - 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 + 3.50T + 61T^{2} \) |
| 67 | \( 1 - 12.7iT - 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 0.0111iT - 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 2.64iT - 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−9.126357771938035008120442348661, −8.901588565547597800453504743396, −7.47173208545879625589045292853, −6.86021051225909715615914815433, −5.68297631931882013516684817195, −5.03471109490484629857438838902, −4.16292126941723955662025525751, −3.58824144413364086282741380640, −2.05696802685984572610530490511, −0.74721543601711695574083415855,
1.68947733212085388026553443830, 1.87329077680714895846456021922, 3.35491292865003803807588634547, 4.25692192971509517664410903549, 5.76363004717439664253608378716, 6.26544024546337834734547117465, 6.79987430707157667755836450449, 7.62701069748278414787067573994, 8.570982315515976122746064463697, 9.277407761810918041413856622389
