L(s) = 1 | + 2.68·2-s + 5.22·4-s + 1.68·7-s + 8.65·8-s − 1.07·11-s + 2.67·13-s + 4.53·14-s + 12.8·16-s + 3.93·17-s − 1.17·19-s − 2.89·22-s + 4.06·23-s + 7.19·26-s + 8.80·28-s − 5.95·29-s − 7.10·31-s + 17.1·32-s + 10.5·34-s − 4.58·37-s − 3.16·38-s + 11.2·41-s + 2.58·43-s − 5.63·44-s + 10.9·46-s + 1.91·47-s − 4.15·49-s + 13.9·52-s + ⋯ |
L(s) = 1 | + 1.90·2-s + 2.61·4-s + 0.637·7-s + 3.05·8-s − 0.325·11-s + 0.742·13-s + 1.21·14-s + 3.20·16-s + 0.953·17-s − 0.270·19-s − 0.618·22-s + 0.847·23-s + 1.41·26-s + 1.66·28-s − 1.10·29-s − 1.27·31-s + 3.02·32-s + 1.81·34-s − 0.753·37-s − 0.513·38-s + 1.75·41-s + 0.394·43-s − 0.849·44-s + 1.60·46-s + 0.279·47-s − 0.593·49-s + 1.93·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.306266119\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.306266119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 + 5.95T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 - 1.91T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 + 1.33T + 59T^{2} \) |
| 61 | \( 1 + 7.28T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 5.98T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 4.00T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.70658304771929489521804248359, −7.32347956035807659264307256503, −6.45747906199336694442240388264, −5.60643501275079438032587739126, −5.37719730344467394649349633683, −4.45374383117868646283535614147, −3.78126267251125159436491386682, −3.11563733635217610049025414612, −2.17162573620598780266860342356, −1.30889348925873739567633997416,
1.30889348925873739567633997416, 2.17162573620598780266860342356, 3.11563733635217610049025414612, 3.78126267251125159436491386682, 4.45374383117868646283535614147, 5.37719730344467394649349633683, 5.60643501275079438032587739126, 6.45747906199336694442240388264, 7.32347956035807659264307256503, 7.70658304771929489521804248359