| L(s) = 1 | + 6.87e10·2-s + 2.83e17·3-s + 4.72e21·4-s + 1.95e28·6-s + 3.24e32·8-s + 5.79e34·9-s + 5.90e37·11-s + 1.34e39·12-s + 2.23e43·16-s − 2.06e44·17-s + 3.98e45·18-s − 1.24e46·19-s + 4.05e48·22-s + 9.20e49·24-s + 2.11e50·25-s + 1.00e52·27-s + 1.53e54·32-s + 1.67e55·33-s − 1.41e55·34-s + 2.73e56·36-s − 8.56e56·38-s − 1.95e58·41-s + 1.17e59·43-s + 2.78e59·44-s + 6.32e60·48-s + 7.03e60·49-s + 1.45e61·50-s + ⋯ |
| L(s) = 1 | + 2-s + 1.89·3-s + 4-s + 1.89·6-s + 8-s + 2.57·9-s + 1.91·11-s + 1.89·12-s + 16-s − 1.04·17-s + 2.57·18-s − 1.14·19-s + 1.91·22-s + 1.89·24-s + 25-s + 2.97·27-s + 32-s + 3.61·33-s − 1.04·34-s + 2.57·36-s − 1.14·38-s − 1.70·41-s + 1.84·43-s + 1.91·44-s + 1.89·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(73-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+36) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{73}{2})\) |
\(\approx\) |
\(13.08455643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.08455643\) |
| \(L(37)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{36} T \) |
| good | 3 | \( 1 - 283772970975307042 T + p^{72} T^{2} \) |
| 5 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 7 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 11 | \( 1 - \)\(59\!\cdots\!22\)\( T + p^{72} T^{2} \) |
| 13 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 17 | \( 1 + \)\(20\!\cdots\!38\)\( T + p^{72} T^{2} \) |
| 19 | \( 1 + \)\(12\!\cdots\!38\)\( T + p^{72} T^{2} \) |
| 23 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 29 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 31 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 37 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 41 | \( 1 + \)\(19\!\cdots\!18\)\( T + p^{72} T^{2} \) |
| 43 | \( 1 - \)\(11\!\cdots\!02\)\( T + p^{72} T^{2} \) |
| 47 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 53 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 59 | \( 1 + \)\(91\!\cdots\!18\)\( T + p^{72} T^{2} \) |
| 61 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 67 | \( 1 - \)\(41\!\cdots\!62\)\( T + p^{72} T^{2} \) |
| 71 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 73 | \( 1 + \)\(13\!\cdots\!78\)\( T + p^{72} T^{2} \) |
| 79 | \( ( 1 - p^{36} T )( 1 + p^{36} T ) \) |
| 83 | \( 1 - \)\(54\!\cdots\!62\)\( T + p^{72} T^{2} \) |
| 89 | \( 1 + \)\(30\!\cdots\!78\)\( T + p^{72} T^{2} \) |
| 97 | \( 1 - \)\(53\!\cdots\!42\)\( T + p^{72} T^{2} \) |
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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
−10.68183492442552101262544847904, −9.253544125488329959658537533512, −8.508138074251920619694847353096, −7.15297040456880723750694280501, −6.44175792681326882373123471404, −4.44803448786479317231391386699, −3.95247376198352848095940045632, −2.95477328251242472896636367097, −2.03690760503799990776713689906, −1.26873379467242487320456220006,
1.26873379467242487320456220006, 2.03690760503799990776713689906, 2.95477328251242472896636367097, 3.95247376198352848095940045632, 4.44803448786479317231391386699, 6.44175792681326882373123471404, 7.15297040456880723750694280501, 8.508138074251920619694847353096, 9.253544125488329959658537533512, 10.68183492442552101262544847904
