L(s) = 1 | + 0.414·2-s − 1.82·4-s + 0.828·7-s − 1.58·8-s − 0.585·11-s + 13-s + 0.343·14-s + 3·16-s − 4.82·17-s + 3.41·19-s − 0.242·22-s − 1.41·23-s + 0.414·26-s − 1.51·28-s − 5.65·29-s + 10.2·31-s + 4.41·32-s − 1.99·34-s − 8.48·37-s + 1.41·38-s + 8.82·41-s − 3.07·43-s + 1.07·44-s − 0.585·46-s + 0.828·47-s − 6.31·49-s − 1.82·52-s + ⋯ |
L(s) = 1 | + 0.292·2-s − 0.914·4-s + 0.313·7-s − 0.560·8-s − 0.176·11-s + 0.277·13-s + 0.0917·14-s + 0.750·16-s − 1.17·17-s + 0.783·19-s − 0.0517·22-s − 0.294·23-s + 0.0812·26-s − 0.286·28-s − 1.05·29-s + 1.83·31-s + 0.780·32-s − 0.342·34-s − 1.39·37-s + 0.229·38-s + 1.37·41-s − 0.468·43-s + 0.161·44-s − 0.0863·46-s + 0.120·47-s − 0.901·49-s − 0.253·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 + 0.585T + 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 + 3.07T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.275731186737373112759147165152, −7.889392780197851197501579569338, −6.74903189495447675729284921469, −6.00436687902283535343009165114, −5.10917826261536785364567787512, −4.54358292044570348135193158744, −3.69942047285066322199294326367, −2.76739397170308285048990857327, −1.44217323418309889110440269678, 0,
1.44217323418309889110440269678, 2.76739397170308285048990857327, 3.69942047285066322199294326367, 4.54358292044570348135193158744, 5.10917826261536785364567787512, 6.00436687902283535343009165114, 6.74903189495447675729284921469, 7.889392780197851197501579569338, 8.275731186737373112759147165152