L(s) = 1 | − 2·5-s − 4·13-s + 8·17-s − 25-s + 10·29-s − 12·37-s − 8·41-s − 7·49-s − 14·53-s − 12·61-s + 8·65-s + 6·73-s − 16·85-s − 16·89-s + 18·97-s + 2·101-s − 20·109-s − 16·113-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 1.10·13-s + 1.94·17-s − 1/5·25-s + 1.85·29-s − 1.97·37-s − 1.24·41-s − 49-s − 1.92·53-s − 1.53·61-s + 0.992·65-s + 0.702·73-s − 1.73·85-s − 1.69·89-s + 1.82·97-s + 0.199·101-s − 1.91·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.374317050746941678652045723724, −7.905649550307471818217312509867, −7.20985741162712884886550996326, −6.38783730439590999755202593371, −5.26281331067729359138909959698, −4.70249568691178473950725828700, −3.56660939918771924244691532023, −2.94387784296716633125559315613, −1.48150632275528664287842078808, 0,
1.48150632275528664287842078808, 2.94387784296716633125559315613, 3.56660939918771924244691532023, 4.70249568691178473950725828700, 5.26281331067729359138909959698, 6.38783730439590999755202593371, 7.20985741162712884886550996326, 7.905649550307471818217312509867, 8.374317050746941678652045723724