| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 3.80·7-s + 8-s + 9-s − 6.13·11-s − 12-s − 4.13·13-s − 3.80·14-s + 16-s + 6.80·17-s + 18-s − 7.60·19-s + 3.80·21-s − 6.13·22-s + 2·23-s − 24-s − 4.13·26-s − 27-s − 3.80·28-s + 6.80·29-s − 31-s + 32-s + 6.13·33-s + 6.80·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.43·7-s + 0.353·8-s + 0.333·9-s − 1.85·11-s − 0.288·12-s − 1.14·13-s − 1.01·14-s + 0.250·16-s + 1.65·17-s + 0.235·18-s − 1.74·19-s + 0.830·21-s − 1.30·22-s + 0.417·23-s − 0.204·24-s − 0.811·26-s − 0.192·27-s − 0.718·28-s + 1.26·29-s − 0.179·31-s + 0.176·32-s + 1.06·33-s + 1.16·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224731679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224731679\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 + 6.13T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 - 6.80T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 37 | \( 1 + 8.13T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 3.33T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + 8.27T + 59T^{2} \) |
| 61 | \( 1 + 4.13T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.056821129182108923787759866792, −7.40115082944199664297898212573, −6.74974753704165953930396993384, −5.99592314284602498793469529539, −5.37258876743113723929884765723, −4.78010204163407449248737534927, −3.76889656850593026516049765991, −2.88070683260513755142109517390, −2.32035493249962011122116839537, −0.52924925277956732473628264734,
0.52924925277956732473628264734, 2.32035493249962011122116839537, 2.88070683260513755142109517390, 3.76889656850593026516049765991, 4.78010204163407449248737534927, 5.37258876743113723929884765723, 5.99592314284602498793469529539, 6.74974753704165953930396993384, 7.40115082944199664297898212573, 8.056821129182108923787759866792
