L(s) = 1 | + 2.64·2-s + 5.00·4-s + 3.64·7-s + 7.93·8-s − 5.64·11-s + 5.64·13-s + 9.64·14-s + 11.0·16-s − 4·17-s − 19-s − 14.9·22-s − 1.29·23-s + 14.9·26-s + 18.2·28-s + 6.93·29-s + 6·31-s + 13.2·32-s − 10.5·34-s + 1.64·37-s − 2.64·38-s + 4.35·41-s − 0.354·43-s − 28.2·44-s − 3.41·46-s + 9.29·47-s + 6.29·49-s + 28.2·52-s + ⋯ |
L(s) = 1 | + 1.87·2-s + 2.50·4-s + 1.37·7-s + 2.80·8-s − 1.70·11-s + 1.56·13-s + 2.57·14-s + 2.75·16-s − 0.970·17-s − 0.229·19-s − 3.18·22-s − 0.269·23-s + 2.92·26-s + 3.44·28-s + 1.28·29-s + 1.07·31-s + 2.33·32-s − 1.81·34-s + 0.270·37-s − 0.429·38-s + 0.680·41-s − 0.0540·43-s − 4.25·44-s − 0.503·46-s + 1.35·47-s + 0.898·49-s + 3.91·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.665106242\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.665106242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 - 5.64T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 + 1.29T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 4.35T + 41T^{2} \) |
| 43 | \( 1 + 0.354T + 43T^{2} \) |
| 47 | \( 1 - 9.29T + 47T^{2} \) |
| 53 | \( 1 - 0.708T + 53T^{2} \) |
| 59 | \( 1 + 0.708T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.135411688254788252390039068312, −7.56215344202402602209250687593, −6.61778941410098529639593165243, −5.91023780505172665245629398562, −5.33448826175436804155438190400, −4.49752532319727424406536205083, −4.21949447898633211738124960599, −2.95475010740437212624032828097, −2.38066128886388918710836610739, −1.34721384391766196296776301389,
1.34721384391766196296776301389, 2.38066128886388918710836610739, 2.95475010740437212624032828097, 4.21949447898633211738124960599, 4.49752532319727424406536205083, 5.33448826175436804155438190400, 5.91023780505172665245629398562, 6.61778941410098529639593165243, 7.56215344202402602209250687593, 8.135411688254788252390039068312