L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + 3.41i·11-s + (−4 − 4i)13-s + 1.00·14-s − 1.00·16-s + (−3.41 − 3.41i)17-s + 2.82i·19-s + (2.41 − 2.41i)22-s + (0.828 − 0.828i)23-s + 5.65i·26-s + (−0.707 − 0.707i)28-s − 4.82·29-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + 1.02i·11-s + (−1.10 − 1.10i)13-s + 0.267·14-s − 0.250·16-s + (−0.828 − 0.828i)17-s + 0.648i·19-s + (0.514 − 0.514i)22-s + (0.172 − 0.172i)23-s + 1.10i·26-s + (−0.133 − 0.133i)28-s − 0.896·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.051247715\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.051247715\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 - 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (4 + 4i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.41 + 3.41i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + (-0.828 + 0.828i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + (-2.24 + 2.24i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.82iT - 41T^{2} \) |
| 43 | \( 1 + (-8.07 - 8.07i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.757 + 0.757i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.41 + 5.41i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + (5.58 - 5.58i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.82iT - 71T^{2} \) |
| 73 | \( 1 + (-7.07 - 7.07i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.65iT - 79T^{2} \) |
| 83 | \( 1 + (4.82 - 4.82i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.82T + 89T^{2} \) |
| 97 | \( 1 + (-7.41 + 7.41i)T - 97iT^{2} \) |
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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
−8.800819610362495586150622132263, −7.906110842146838160309692879030, −7.35762168038180591105661024173, −6.61898344630869156134064380585, −5.52811744282305580891745826786, −4.76303431715305975095088606269, −3.89513974211867058115027798370, −2.68227119872374468039129633600, −2.26292775782187156695250846905, −0.72753017720800673085444941335,
0.58102001948101867782320349448, 1.95536680568999105155571947792, 2.96963474987026152988058099722, 4.19048041013594969349771256930, 4.81338630475251227989444935888, 5.93128344784973761588199548937, 6.48965846482341436286266605069, 7.18531369384163984432314008640, 7.937080081456609917677242792860, 8.757701440729275528281114744054