L(s) = 1 | − 3-s − 3.81·5-s − 7-s + 9-s + 4.73·11-s + 13-s + 3.81·15-s − 5.22·17-s − 2.92·19-s + 21-s − 3.33·23-s + 9.55·25-s − 27-s − 0.922·29-s + 7.51·31-s − 4.73·33-s + 3.81·35-s + 0.154·37-s − 39-s + 6.36·41-s + 6.55·43-s − 3.81·45-s − 9.03·47-s + 49-s + 5.22·51-s + 8.55·53-s − 18.0·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.70·5-s − 0.377·7-s + 0.333·9-s + 1.42·11-s + 0.277·13-s + 0.984·15-s − 1.26·17-s − 0.670·19-s + 0.218·21-s − 0.694·23-s + 1.91·25-s − 0.192·27-s − 0.171·29-s + 1.35·31-s − 0.824·33-s + 0.644·35-s + 0.0254·37-s − 0.160·39-s + 0.994·41-s + 0.999·43-s − 0.568·45-s − 1.31·47-s + 0.142·49-s + 0.731·51-s + 1.17·53-s − 2.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 3.81T + 5T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 + 0.922T + 29T^{2} \) |
| 31 | \( 1 - 7.51T + 31T^{2} \) |
| 37 | \( 1 - 0.154T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 - 8.55T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 7.73T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 1.40T + 83T^{2} \) |
| 89 | \( 1 + 1.96T + 89T^{2} \) |
| 97 | \( 1 + 2.11T + 97T^{2} \) |
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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
−8.069911760640105394766209541968, −7.10553704569394816009048973191, −6.64078503298745854384165795732, −6.01828820532158172226743560416, −4.75927787377057189179851769076, −4.05272872141826898148252331157, −3.81646177768785286233772553466, −2.51087080987043957889307459634, −1.06388457822202578151231961597, 0,
1.06388457822202578151231961597, 2.51087080987043957889307459634, 3.81646177768785286233772553466, 4.05272872141826898148252331157, 4.75927787377057189179851769076, 6.01828820532158172226743560416, 6.64078503298745854384165795732, 7.10553704569394816009048973191, 8.069911760640105394766209541968