| L(s) = 1 | + 0.665·2-s − 3-s − 1.55·4-s + 5-s − 0.665·6-s − 2.36·8-s + 9-s + 0.665·10-s + 5.61·11-s + 1.55·12-s − 6.44·13-s − 15-s + 1.54·16-s + 0.947·17-s + 0.665·18-s + 6.91·19-s − 1.55·20-s + 3.73·22-s + 1.53·23-s + 2.36·24-s + 25-s − 4.28·26-s − 27-s + 8.99·29-s − 0.665·30-s − 2.91·31-s + 5.75·32-s + ⋯ |
| L(s) = 1 | + 0.470·2-s − 0.577·3-s − 0.778·4-s + 0.447·5-s − 0.271·6-s − 0.836·8-s + 0.333·9-s + 0.210·10-s + 1.69·11-s + 0.449·12-s − 1.78·13-s − 0.258·15-s + 0.385·16-s + 0.229·17-s + 0.156·18-s + 1.58·19-s − 0.348·20-s + 0.796·22-s + 0.319·23-s + 0.482·24-s + 0.200·25-s − 0.840·26-s − 0.192·27-s + 1.67·29-s − 0.121·30-s − 0.523·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.444237584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444237584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.665T + 2T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 - 0.947T + 17T^{2} \) |
| 19 | \( 1 - 6.91T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 - 8.99T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 - 0.118T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 + 5.48T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 + 8.72T + 67T^{2} \) |
| 71 | \( 1 + 1.44T + 71T^{2} \) |
| 73 | \( 1 - 7.78T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.17T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 - 7.93T + 97T^{2} \) |
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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
−10.15002417988774757054114754445, −9.553711254666649423151352525948, −8.997401902438930073329350775033, −7.62015610013177773132400259921, −6.71111772001510652676134277245, −5.74908963562171304872962707513, −4.94856159933190019232139384814, −4.16454559340968496211215467140, −2.88425092757040816850105035440, −1.02887285682465642704345923603,
1.02887285682465642704345923603, 2.88425092757040816850105035440, 4.16454559340968496211215467140, 4.94856159933190019232139384814, 5.74908963562171304872962707513, 6.71111772001510652676134277245, 7.62015610013177773132400259921, 8.997401902438930073329350775033, 9.553711254666649423151352525948, 10.15002417988774757054114754445
