| L(s) = 1 | − 4.51e5·3-s + 1.67e7·4-s + 2.44e8·5-s + 1.38e10·7-s − 7.87e10·9-s + 1.66e12·11-s − 7.57e12·12-s − 4.27e13·13-s − 1.10e14·15-s + 2.81e14·16-s + 1.11e15·17-s + 4.09e15·20-s − 6.24e15·21-s + 5.96e16·25-s + 1.63e17·27-s + 2.32e17·28-s + 1.27e17·29-s − 7.49e17·33-s + 3.37e18·35-s − 1.32e18·36-s + 1.93e19·39-s + 2.78e19·44-s − 1.92e19·45-s − 1.46e20·47-s − 1.27e20·48-s + 1.91e20·49-s − 5.01e20·51-s + ⋯ |
| L(s) = 1 | − 0.849·3-s + 4-s + 5-s + 7-s − 0.278·9-s + 0.529·11-s − 0.849·12-s − 1.83·13-s − 0.849·15-s + 16-s + 1.90·17-s + 20-s − 0.849·21-s + 25-s + 1.08·27-s + 28-s + 0.359·29-s − 0.449·33-s + 35-s − 0.278·36-s + 1.55·39-s + 0.529·44-s − 0.278·45-s − 1.25·47-s − 0.849·48-s + 49-s − 1.61·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{25}{2})\) |
\(\approx\) |
\(3.159104818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.159104818\) |
| \(L(13)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{12} T \) |
| 7 | \( 1 - p^{12} T \) |
| good | 2 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 3 | \( 1 + 451358 T + p^{24} T^{2} \) |
| 11 | \( 1 - 1660671885602 T + p^{24} T^{2} \) |
| 13 | \( 1 + 42760283479198 T + p^{24} T^{2} \) |
| 17 | \( 1 - 1110677308191362 T + p^{24} T^{2} \) |
| 19 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 23 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 29 | \( 1 - 127175001160727522 T + p^{24} T^{2} \) |
| 31 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 37 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 41 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 43 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 47 | \( 1 + \)\(14\!\cdots\!18\)\( T + p^{24} T^{2} \) |
| 53 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 59 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 61 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 67 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 71 | \( 1 - \)\(32\!\cdots\!22\)\( T + p^{24} T^{2} \) |
| 73 | \( 1 + \)\(43\!\cdots\!98\)\( T + p^{24} T^{2} \) |
| 79 | \( 1 - \)\(25\!\cdots\!42\)\( T + p^{24} T^{2} \) |
| 83 | \( 1 - \)\(20\!\cdots\!82\)\( T + p^{24} T^{2} \) |
| 89 | \( ( 1 - p^{12} T )( 1 + p^{12} T ) \) |
| 97 | \( 1 + \)\(64\!\cdots\!58\)\( T + p^{24} T^{2} \) |
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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
−11.78513976050527971804131463982, −10.63136250765106274600623634306, −9.721319863544906375541025876711, −7.936417132314673283069095035630, −6.79854449753749513131263413192, −5.66273282382362333295148459696, −5.00118972050492590488134815372, −2.91683879342237463108478478400, −1.85501652612947094003150559660, −0.881604789163205340557431765603,
0.881604789163205340557431765603, 1.85501652612947094003150559660, 2.91683879342237463108478478400, 5.00118972050492590488134815372, 5.66273282382362333295148459696, 6.79854449753749513131263413192, 7.936417132314673283069095035630, 9.721319863544906375541025876711, 10.63136250765106274600623634306, 11.78513976050527971804131463982
