L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s − 3·8-s + 9-s − 10-s − 2·12-s + 4·13-s − 2·15-s − 16-s − 4·17-s + 18-s − 8·19-s + 20-s − 6·24-s + 25-s + 4·26-s − 4·27-s + 6·29-s − 2·30-s + 6·31-s + 5·32-s − 4·34-s − 36-s − 6·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s + 1.10·13-s − 0.516·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 1.22·24-s + 1/5·25-s + 0.784·26-s − 0.769·27-s + 1.11·29-s − 0.365·30-s + 1.07·31-s + 0.883·32-s − 0.685·34-s − 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.31154383598114, −14.71596511960727, −14.36952550725841, −13.82643685133173, −13.33129205384708, −13.01291797795676, −12.49470082977256, −11.65681767263396, −11.40004030581798, −10.35297670259613, −10.21951252375975, −9.112694357894602, −8.675197800592340, −8.567867210414560, −8.053759835846941, −7.051721061602976, −6.527791788358088, −5.929554263078207, −5.195196106872050, −4.329701444267564, −4.036022956456469, −3.572872559445093, −2.610097560227320, −2.360496133550627, −1.076051285792407, 0,
1.076051285792407, 2.360496133550627, 2.610097560227320, 3.572872559445093, 4.036022956456469, 4.329701444267564, 5.195196106872050, 5.929554263078207, 6.527791788358088, 7.051721061602976, 8.053759835846941, 8.567867210414560, 8.675197800592340, 9.112694357894602, 10.21951252375975, 10.35297670259613, 11.40004030581798, 11.65681767263396, 12.49470082977256, 13.01291797795676, 13.33129205384708, 13.82643685133173, 14.36952550725841, 14.71596511960727, 15.31154383598114