L(s) = 1 | + (−0.610 − 0.0966i)2-s + (−0.587 − 0.190i)4-s + (−0.891 + 0.453i)7-s + (0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (−0.809 − 0.587i)11-s + (0.587 − 0.190i)14-s + (0.280 + 0.550i)18-s + (0.437 + 0.437i)22-s + (1.34 + 1.34i)23-s + (0.610 − 0.0966i)28-s + (−0.363 + 1.11i)29-s + (−0.707 − 0.707i)32-s + (0.190 + 0.587i)36-s + (0.863 + 1.69i)37-s + ⋯ |
L(s) = 1 | + (−0.610 − 0.0966i)2-s + (−0.587 − 0.190i)4-s + (−0.891 + 0.453i)7-s + (0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (−0.809 − 0.587i)11-s + (0.587 − 0.190i)14-s + (0.280 + 0.550i)18-s + (0.437 + 0.437i)22-s + (1.34 + 1.34i)23-s + (0.610 − 0.0966i)28-s + (−0.363 + 1.11i)29-s + (−0.707 − 0.707i)32-s + (0.190 + 0.587i)36-s + (0.863 + 1.69i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3599549390\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3599549390\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (0.891 - 0.453i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (0.610 + 0.0966i)T + (0.951 + 0.309i)T^{2} \) |
| 3 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 13 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.34 - 1.34i)T + iT^{2} \) |
| 29 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.863 - 1.69i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + (1.14 - 1.14i)T - iT^{2} \) |
| 47 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (1.16 + 0.183i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.831 - 0.831i)T - iT^{2} \) |
| 71 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.951 + 0.309i)T^{2} \) |
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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
−9.607215188501490530828669291452, −8.801748157107014375859397855190, −8.337422491097386798096923120559, −7.32847102367528182692778088599, −6.36778819487240129258938674626, −5.55855826543431378279005114195, −4.87856936717468442311399743903, −3.47372277478618093175476287492, −2.89690355618453304309813962752, −1.20732778528109601876311556512,
0.36622070047022712646765634837, 2.22776049909773345645613158366, 3.27052711034396073534136605642, 4.38929484749259942848587886952, 5.05143533214386780399732484633, 6.10718517956591315864772190750, 7.16934916111430154650508667079, 7.68133463496012051151796162099, 8.479065059966077480084221003678, 9.199616318619507195315356206136