| L(s) = 1 | + i·3-s − 4i·7-s − 9-s − 4·11-s + 6i·13-s − 2i·17-s + 4·19-s + 4·21-s − i·27-s − 10·29-s + 4·31-s − 4i·33-s + 10i·37-s − 6·39-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.51i·7-s − 0.333·9-s − 1.20·11-s + 1.66i·13-s − 0.485i·17-s + 0.917·19-s + 0.872·21-s − 0.192i·27-s − 1.85·29-s + 0.718·31-s − 0.696i·33-s + 1.64i·37-s − 0.960·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9364448527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9364448527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.479786297725379557401367451644, −8.431167674893431240273158149556, −7.52208453627361906068871526447, −7.12045358834248287670667576442, −6.10678740453968074165149771714, −5.03863595766506814511239974879, −4.42750209507042207071434516036, −3.66598083613777348574004694925, −2.64074576759699775520614342692, −1.23974259650489176352057723804,
0.32440630939638020937516136482, 1.98265859570720987021060056416, 2.72045696038034517593428814015, 3.55372316540734343510354459001, 5.30910556846376591302458578515, 5.41176320910392119723875010149, 6.17716887028081262497611723478, 7.47692408071013330466377165782, 7.84432273651876567989963926553, 8.630321927183860793668633828208
