| L(s) = 1 | − 1.73·3-s + 2.82i·5-s + 4.89i·7-s + 2.99·9-s − 3.46·11-s − 4.89i·15-s − 8.48i·21-s − 3.00·25-s − 5.19·27-s − 2.82i·29-s − 4.89i·31-s + 5.99·33-s − 13.8·35-s + 8.48i·45-s − 16.9·49-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 1.26i·5-s + 1.85i·7-s + 0.999·9-s − 1.04·11-s − 1.26i·15-s − 1.85i·21-s − 0.600·25-s − 1.00·27-s − 0.525i·29-s − 0.879i·31-s + 1.04·33-s − 2.34·35-s + 1.26i·45-s − 2.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.621190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.621190i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 4.89iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14.1iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.85706137974362881450179570029, −10.07119714270436799817653315049, −9.217481215712470752668141570010, −8.042184436513925800335542659949, −7.16711275172875094805076375514, −6.07465737404072177471866593433, −5.74305601871394028333282368187, −4.64500702455603336864548345378, −3.03813145797375299224413978357, −2.18484952672149358638869733291,
0.36826834496503402790335027340, 1.42884296640537046417987972835, 3.64304143512017664889994801086, 4.72500145604183245889614597180, 5.09143092347430032188987251817, 6.37453156957916658313728768998, 7.32570669545723150595181355179, 7.981567038197500530863098914061, 9.144678046671892915320435899698, 10.26417275964654300059175735370
