| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 11-s + 4·13-s + 2·14-s + 16-s + 17-s − 6·19-s + 22-s − 2·23-s + 4·26-s + 2·28-s + 6·29-s − 4·31-s + 32-s + 34-s + 2·37-s − 6·38-s + 6·41-s − 6·43-s + 44-s − 2·46-s − 12·47-s − 3·49-s + 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.301·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 1.37·19-s + 0.213·22-s − 0.417·23-s + 0.784·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.328·37-s − 0.973·38-s + 0.937·41-s − 0.914·43-s + 0.150·44-s − 0.294·46-s − 1.75·47-s − 3/7·49-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.29295201097118, −13.70379943286772, −13.16615880126981, −12.82317765262490, −12.23244696554918, −11.71380711713105, −11.27335926792436, −10.79435634906550, −10.45621965634577, −9.723932929375354, −9.145004839626094, −8.446704554725017, −8.150242635228485, −7.697095091472498, −6.735572149999055, −6.520596172721486, −5.965377607340219, −5.331100643542048, −4.756907281060262, −4.223968219878709, −3.767358701264011, −3.088641817541762, −2.380793552864099, −1.637402159045409, −1.215191295133865, 0,
1.215191295133865, 1.637402159045409, 2.380793552864099, 3.088641817541762, 3.767358701264011, 4.223968219878709, 4.756907281060262, 5.331100643542048, 5.965377607340219, 6.520596172721486, 6.735572149999055, 7.697095091472498, 8.150242635228485, 8.446704554725017, 9.145004839626094, 9.723932929375354, 10.45621965634577, 10.79435634906550, 11.27335926792436, 11.71380711713105, 12.23244696554918, 12.82317765262490, 13.16615880126981, 13.70379943286772, 14.29295201097118
