L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s − 3·11-s − 13-s + 16-s + 3·17-s − 7·19-s + 3·20-s + 3·22-s − 9·23-s + 4·25-s + 26-s + 3·29-s + 8·31-s − 32-s − 3·34-s − 37-s + 7·38-s − 3·40-s + 3·41-s − 43-s − 3·44-s + 9·46-s − 4·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s − 0.904·11-s − 0.277·13-s + 1/4·16-s + 0.727·17-s − 1.60·19-s + 0.670·20-s + 0.639·22-s − 1.87·23-s + 4/5·25-s + 0.196·26-s + 0.557·29-s + 1.43·31-s − 0.176·32-s − 0.514·34-s − 0.164·37-s + 1.13·38-s − 0.474·40-s + 0.468·41-s − 0.152·43-s − 0.452·44-s + 1.32·46-s − 0.565·50-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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.69116398365528820620013768500, −6.69075382137987652841918976260, −6.16891289969289323136774994723, −5.63569446220630569032511057343, −4.81753442605953036443568506567, −3.87613157450784483134030052236, −2.58297218811614095443441327119, −2.30076310590399468098089112386, −1.32148331949439410293917292003, 0,
1.32148331949439410293917292003, 2.30076310590399468098089112386, 2.58297218811614095443441327119, 3.87613157450784483134030052236, 4.81753442605953036443568506567, 5.63569446220630569032511057343, 6.16891289969289323136774994723, 6.69075382137987652841918976260, 7.69116398365528820620013768500