L(s) = 1 | + (−0.852 + 0.522i)3-s + (0.987 + 0.156i)7-s + (0.453 − 0.891i)9-s + (0.0784 − 0.996i)13-s + (−0.309 + 0.951i)17-s + (0.852 − 0.522i)19-s + (−0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (0.0784 + 0.996i)27-s + (0.972 − 0.233i)29-s + (0.309 + 0.951i)31-s + (−0.522 + 0.852i)37-s + (0.453 + 0.891i)39-s + (0.156 + 0.987i)41-s + (−0.382 − 0.923i)43-s + ⋯ |
L(s) = 1 | + (−0.852 + 0.522i)3-s + (0.987 + 0.156i)7-s + (0.453 − 0.891i)9-s + (0.0784 − 0.996i)13-s + (−0.309 + 0.951i)17-s + (0.852 − 0.522i)19-s + (−0.923 + 0.382i)21-s + (0.707 − 0.707i)23-s + (0.0784 + 0.996i)27-s + (0.972 − 0.233i)29-s + (0.309 + 0.951i)31-s + (−0.522 + 0.852i)37-s + (0.453 + 0.891i)39-s + (0.156 + 0.987i)41-s + (−0.382 − 0.923i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526796066 + 0.2480338946i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526796066 + 0.2480338946i\) |
\(L(1)\) |
\(\approx\) |
\(0.9860715817 + 0.1187914061i\) |
\(L(1)\) |
\(\approx\) |
\(0.9860715817 + 0.1187914061i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.852 + 0.522i)T \) |
| 7 | \( 1 + (0.987 + 0.156i)T \) |
| 13 | \( 1 + (0.0784 - 0.996i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.852 - 0.522i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
| 29 | \( 1 + (0.972 - 0.233i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.522 + 0.852i)T \) |
| 41 | \( 1 + (0.156 + 0.987i)T \) |
| 43 | \( 1 + (-0.382 - 0.923i)T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.760 - 0.649i)T \) |
| 59 | \( 1 + (-0.852 - 0.522i)T \) |
| 61 | \( 1 + (-0.649 + 0.760i)T \) |
| 67 | \( 1 + (-0.382 + 0.923i)T \) |
| 71 | \( 1 + (0.453 + 0.891i)T \) |
| 73 | \( 1 + (-0.156 + 0.987i)T \) |
| 79 | \( 1 + (0.951 - 0.309i)T \) |
| 83 | \( 1 + (-0.649 + 0.760i)T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.38895218727799595592652834690, −18.14699386666291726480569442210, −17.33783897219729906717764276694, −16.77966197222162060398882395383, −16.07359400222180757205844384559, −15.40901977162419261033083826077, −14.33039518109704589451571946598, −13.84941850655461991500436324477, −13.26245703360881789818183786864, −12.12089269170768979292011858968, −11.83553538900247696162013273180, −11.13007508642533442871819538247, −10.5828608976699383826440630749, −9.551304804395668663439402475211, −8.88656167568126336921455582093, −7.73255309159702537556770457753, −7.468748430026982310714651878348, −6.5787938705381543741813689068, −5.84914790421591190650889131721, −4.89369422114641619836276109780, −4.64487550056417026838054529025, −3.468568065660494632412999697563, −2.28590207979572914632050506300, −1.5654168191910060900725818074, −0.77354590306733137559267312215,
0.77017770357041003547353869541, 1.52588078936137061166915050425, 2.775228433464619961227934510390, 3.5738937794749735984077902876, 4.66047221779956144743346503685, 4.977221044299357104562551584637, 5.77175128972526900148738403465, 6.57047743598075259197678866606, 7.324369092371059209544726824182, 8.40790662424207499574594815392, 8.721173428451947247709582266402, 9.9953664251771975326704395527, 10.36266196581114890148793778965, 11.132941954933075429621449633970, 11.68831299802547707996443463036, 12.40403209368093898795884482711, 13.110828037736376183384036316124, 14.014167929830012773911530609732, 14.90576340300952674335245695048, 15.319885716063191622948888701946, 15.95804742010881445043600997998, 16.87759820838750286965953733889, 17.39804129539945627477338217898, 17.98409586693960225022060057852, 18.441877169293147991316369331089