L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s + 13-s + 16-s − 2·17-s + 18-s − 19-s + 4·22-s − 23-s + 24-s + 26-s + 27-s + 9·29-s + 32-s + 4·33-s − 2·34-s + 36-s − 2·37-s − 38-s + 39-s + 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.229·19-s + 0.852·22-s − 0.208·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 1.67·29-s + 0.176·32-s + 0.696·33-s − 0.342·34-s + 1/6·36-s − 0.328·37-s − 0.162·38-s + 0.160·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.417486299\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.417486299\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.62673397965624, −12.90944433070267, −12.50453586956133, −12.06174178015751, −11.67408461996282, −10.99373321911716, −10.65040727762041, −10.09918427951152, −9.439247474262894, −9.030427617775508, −8.625530199809919, −7.970104632607917, −7.548111691928862, −6.853621031022454, −6.396152870068434, −6.182783164932568, −5.298262170761159, −4.795646887033986, −4.137257094903105, −3.875017356721015, −3.257831304980631, −2.524127246440809, −2.136632391172048, −1.289942838714494, −0.7313470159378377,
0.7313470159378377, 1.289942838714494, 2.136632391172048, 2.524127246440809, 3.257831304980631, 3.875017356721015, 4.137257094903105, 4.795646887033986, 5.298262170761159, 6.182783164932568, 6.396152870068434, 6.853621031022454, 7.548111691928862, 7.970104632607917, 8.625530199809919, 9.030427617775508, 9.439247474262894, 10.09918427951152, 10.65040727762041, 10.99373321911716, 11.67408461996282, 12.06174178015751, 12.50453586956133, 12.90944433070267, 13.62673397965624