L(s) = 1 | − 2-s + 4-s + 3.44·5-s − 8-s − 3.44·10-s − 2·11-s + 4.89·13-s + 16-s + 2·17-s − 7.44·19-s + 3.44·20-s + 2·22-s + 23-s + 6.89·25-s − 4.89·26-s − 2.89·29-s − 6·31-s − 32-s − 2·34-s − 7.79·37-s + 7.44·38-s − 3.44·40-s − 9.79·41-s − 2.89·43-s − 2·44-s − 46-s − 9.79·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.54·5-s − 0.353·8-s − 1.09·10-s − 0.603·11-s + 1.35·13-s + 0.250·16-s + 0.485·17-s − 1.70·19-s + 0.771·20-s + 0.426·22-s + 0.208·23-s + 1.37·25-s − 0.960·26-s − 0.538·29-s − 1.07·31-s − 0.176·32-s − 0.342·34-s − 1.28·37-s + 1.20·38-s − 0.545·40-s − 1.53·41-s − 0.442·43-s − 0.301·44-s − 0.147·46-s − 1.42·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.44T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 7.44T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 + 9.79T + 41T^{2} \) |
| 43 | \( 1 + 2.89T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 3.10T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 - 7.89T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 6.89T + 97T^{2} \) |
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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
−7.54971107766653956070480491456, −6.65029855042273025578529124128, −6.23311800672535171381941753116, −5.58670672241762335966478457075, −4.92210242290622280731260502350, −3.70816674682945070201229758295, −2.92447412002799934950487610594, −1.82171791204364726840246623034, −1.59004598603295821190051766644, 0,
1.59004598603295821190051766644, 1.82171791204364726840246623034, 2.92447412002799934950487610594, 3.70816674682945070201229758295, 4.92210242290622280731260502350, 5.58670672241762335966478457075, 6.23311800672535171381941753116, 6.65029855042273025578529124128, 7.54971107766653956070480491456