| L(s) = 1 | − 1.33·3-s − 3.16·7-s − 1.21·9-s + 0.0955·11-s − 1.44·13-s + 2.29·17-s − 7.00·19-s + 4.23·21-s − 23-s + 5.63·27-s + 5.39·29-s + 0.584·31-s − 0.127·33-s + 9.29·37-s + 1.92·39-s − 2.86·41-s + 9.50·43-s + 7.09·47-s + 3.03·49-s − 3.07·51-s + 7.73·53-s + 9.36·57-s − 13.6·59-s + 0.234·61-s + 3.84·63-s + 7.49·67-s + 1.33·69-s + ⋯ |
| L(s) = 1 | − 0.771·3-s − 1.19·7-s − 0.404·9-s + 0.0288·11-s − 0.400·13-s + 0.557·17-s − 1.60·19-s + 0.924·21-s − 0.208·23-s + 1.08·27-s + 1.00·29-s + 0.104·31-s − 0.0222·33-s + 1.52·37-s + 0.308·39-s − 0.447·41-s + 1.44·43-s + 1.03·47-s + 0.433·49-s − 0.430·51-s + 1.06·53-s + 1.24·57-s − 1.77·59-s + 0.0300·61-s + 0.483·63-s + 0.915·67-s + 0.160·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 0.0955T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 29 | \( 1 - 5.39T + 29T^{2} \) |
| 31 | \( 1 - 0.584T + 31T^{2} \) |
| 37 | \( 1 - 9.29T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 - 9.50T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 0.234T + 61T^{2} \) |
| 67 | \( 1 - 7.49T + 67T^{2} \) |
| 71 | \( 1 - 5.18T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 - 3.04T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 5.53T + 89T^{2} \) |
| 97 | \( 1 + 2.58T + 97T^{2} \) |
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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
−7.21102041576547423237946843052, −6.47296279198935695347534986303, −6.11502303604048613159460696817, −5.50121936792825214880477002530, −4.58090889309957405369531497326, −3.94866541448336630334087191451, −2.92704727363212899633427416460, −2.38626383282899851846686723086, −0.907997365220015199880931133513, 0,
0.907997365220015199880931133513, 2.38626383282899851846686723086, 2.92704727363212899633427416460, 3.94866541448336630334087191451, 4.58090889309957405369531497326, 5.50121936792825214880477002530, 6.11502303604048613159460696817, 6.47296279198935695347534986303, 7.21102041576547423237946843052
